archimedean copulas in high dimensions: estimators and

Journal de la Société Française de StatistiqueVol. 154 No. 1 (2013)

Archimedean Copulas in High Dimensions:Estimators and Numerical Challenges Motivated by

Financial Applications

Titre: Implémentations stables et estimateurs de copules Archimédiennes en grandes dimensions pour lesapplications financières

Marius Hofert1 , Martin Mächler2 and Alexander J. McNeil3

Abstract: The study of Archimedean dependence models in high dimensions is motivated by current practice inquantitative risk management. The performance of known and new parametric estimators for the parameters ofArchimedean copulas is investigated and related numerical difficulties are addressed. In particular, method-of-moments-like estimators based on pairwise Kendall’s tau, a multivariate extension of Blomqvist’s beta, minimum distanceestimators, the maximum-likelihood estimator, a simulated maximum-likelihood estimator, and a maximum-likelihoodestimator based on the copula diagonal are studied. Their performance is compared in a large-scale simulation studyboth under known and unknown margins (pseudo-observations), in small and high dimensions, under small and largedependencies, and various different Archimedean families. High dimensions up to one hundred are considered andcomputational problems arising from such large dimensions are addressed in detail. All methods are implemented inthe open source R package copula and can thus be easily accessed and studied. The numerical solutions developedin this work extend to various asymmetric generalizations of Archimedean copulas and important quantities such asdistributions of radial parts or the Kendall distribution function.

Résumé : Les pratiques du moment en gestion quantitative du risque nous amènent à étudier des modèles de dépendanceen grandes dimensions construits à partir de copules Archimédiennes. La performance d’un grand nombre d’estimateurs,dont plusieurs sont originaux, est étudiée et des solutions sont proposées pour les problèmes numériques associés. Desestimateurs fondés sur le méthode des moments et le beta de Blomqvist ainsi que le tau de Kendall sont comparés àdes estimateurs du minimum de distance, du maximum de vraisemblance, du maximum de vraisemblance simulé etdu maximum de vraisemblance fondé sur la diagonale de la copule. Des simulations sont réalisées dans le cas où lesmarges sont connues et dans le cas où elles sont estimées non paramétriquement, en faibles et en grandes dimensions,pour plusieurs familles de copules Archimédiennes. Toutes les méthodes étudiées sont implantées dans le packageR copula distribué sous une licence libre. Les solutions numériques présentées dans ce travail restent valident dansle cas de généralisations asymétriques des copules Archimédiennes et d’importantes quantités associées comme lafonction de distribution de Kendall.

Keywords: Archimedean copulas, parameter estimation, Kendall’s tau, Blomqvist’s beta, minimum distance estimators,(diagonal/simulated) maximum-likelihood estimation, quantitative risk managementMots-clés : Copules Archimédiennes, estimation paramétrique, tau de Kendall, beta de Blomqvist, estimateurs duminimum de distance, estimateurs du maximum de vraisemblance, gestion quantitative du risqueAMS 2000 subject classifications: 62H12, 62F10, 62H99, 62H20, 65C60

1 RiskLab, Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland. E-mail:[email protected]. The author (Willis Research Fellow) thanks Willis Re for financial sup-port while this work was being completed.

2 Seminar für Statistik, ETH Zurich, 8092 Zurich, Switzerland. E-mail: [email protected] Department of Actuarial Mathematics and Statistics, Heriot-Watt University, Edinburgh, EH14 4AS, Scotland, and

Maxwell Institute for Mathematical Sciences. E-mail: [email protected].

Journal de la Société Française de Statistique, Vol. 154 No. 1 25-63http://www.sfds.asso.fr/journal

© Société Française de Statistique et Société Mathématique de France (2013) ISSN: 2102-6238

mailto:[email protected]



26 M. Hofert, M. Mächler, A. J. McNeil

1. Introduction

A copula is a multivariate distribution function with standard uniform univariate margins. Animportant class of copulas, known as Archimedean copulas, is given by

C(uuu) = ψ(ψ−1(u1)+ · · ·+ψ−1(ud)), uuu ∈ [0,1]d ,

with generator ψ . As a consequence of this functional form, Archimedean copulas are exchange-able, that is, unaffected by permutations of u1, . . . ,ud . In practical applications, ψ belongs toa parametric family (ψθθθ )θθθ∈Θ whose parameter vector θθθ needs to be estimated; for most popu-lar parametric Archimedean copula families, θθθ is one-dimensional. The aims of this paper aretwo-fold:

1. To carry out a large-scale comparative study of many estimators for Archimedean copulas(including new ones), both under known and unknown margins (pseudo-observations);

2. To focus on the performance of estimators and computations in high dimensions, whereconsiderable numerical challenges have to be overcome.

1.1. Motivation and examples

In high dimensions, none of the existing copula models typically fits data well. Nevertheless, inareas such as the modeling of financial and insurance risks, copula models are applied in highdimensions, often, in our experience, in dimensions that run in to the thousands. As an example,consider a portfolio of d financial instruments or assets with values St,1, . . . ,St,d at time point t,interpreted as today. These asset values are referred to as risk factors and the value of the portfoliois

Vt =d

∑j=1

β jSt, j,

where β j denotes the number of units of asset j in the portfolio, j ∈ {1, . . . ,d}. The asset log-returns are given by

Xt+1, j = log(St+1, j)− log(St, j) = log(St+1, j/St, j), j ∈ {1, . . . ,d},

and the one-period ahead (sign adjusted) loss L of the portfolio is therefore

L =−(Vt+1−Vt) =−d

∑j=1

β j(St+1, j−St, j) =−d

∑j=1

β jSt, j(exp(Xt+1, j)−1). (1)

From a risk management perspective in the context of Basel II/III one is interested in the distri-bution of the loss L. As becomes clear from (1), the loss distribution only depends on the joint,and typically high-dimensional, distribution of the log-returns, all other quantities being known attime point t, that is, today.

Financial portfolios can consist of complex instruments, for example, options, futures, creditdefault swaps and insurance policies. In such cases, the formalism in (1) has to be extended to



Archimedean Copulas in High Dimensions: Estimators and Numerical Challenges 27

incorporate changes in other risk factors, such as price volatilities, default indicators and mortalityrates, for which less information is typically available than for simple asset prices. For computingrisk measures such as Value-at-Risk or expected shortfall, modeling dependence between therisk-factor changes is crucial, as various examples in [McNeil et al., 2005] show.

Dimension reduction techniques cannot always be used in this context since it may be importantto model future payments from each contract or policy. In the absence of more detailed information,exchangeability of the dependence structure is often assumed which makes Archimedean copulascandidate models.

The reason for this is not that these models provide a good fit to the data at hand. Rather, theyare

1. comparably tractable from a numerical and computational point of view in high dimensions;

2. more plausible than models that assume independence and capable of giving indicativeresults on the impact of dependence.

Model misspecification is often accepted in these applications in return for this convenience.Another area of applications where exchangeable dependence models are used is sparse data.

This includes applications to operational risk modeling; see [Chavez-Demoulin et al., 2013] andreferences therein. If the tail is of major interest, Archimedean copulas such as the Clayton orGumbel copula are often considered. They provide alternatives to the t copula (note: the Gaussiancopula does not allow for tail dependence, see [Sibuya, 1959]) to model tail dependence and arealso used for stress testing. A correct specification of the body of the distribution is often nottoo important. One such example is [Schönbucher and Schubert, 2001] and [Hofert and Scherer,2011] in the context of high-dimensional credit-risk portfolios, where copulas are calibratedto essentially five data points only. Nevertheless, the dependence between the underlying 125components in the portfolio plays a crucial role in pricing these products, which has also beenlearned from the recent crisis; see, for example, [Donnelly and Embrechts, 2010]. As shown in[Hofert and Scherer, 2011], even a highly exchangeable model with only one or two parameterscan provide a much more accurate pricing model for such portfolio products. More parametersmay not be justifiable in such applications given the small amount of data available and also notcomputationally feasible since each evaluation of the object function is based on a Monte Carlosimulation. From a practical point of view, one-parameter models are often easy to interpret dueto a one-to-one correspondence with measures of association such as Kendall’s tau. This hascontributed to their popularity; see [Embrechts and Hofert, 2011] for a discussion.

Other examples not detailed here can be found in the area of survival analysis, see [Hougaard,2000] and [Duchateau and Janssen, 2008].

Because they are used in practice in high dimensions, it is important to investigate Archimedeancopulas in that context. Our paper [Hofert et al., 2012] was a first theoretical step in this direction.The present paper complements this work in a statistical and computational direction, detailedbelow. The results extend beyond Archimedean copulas and should be viewed in a wider context.

As the following examples show, the need for accurately evaluating (or estimating) the deriva-tives of generators of Archimedean copulas arises in a variety of contexts.

– The density of a (possibly asymmetric, that is, non-exchangeable) Khoudraji-transformed




Archimedean copula is given by

c(uuu) = ∑J⊆{1,...,d}

ψ(|J|)( d

∑j=1

ψ−1(uα j

j )

)∏j∈J

α j(ψ−1)′(uα j

j )∏j/∈J

(1−α j)u−α jj ,

see [Hofert et al., 2012].– [Hofert, 2010, pp. 117] presents ways to construct Archimedean copulas with more than

one parameter. One example are outer power Archimedean copulas (with generator ψ(t) =ψ(t1/β ), β ≥ 1). As Archimedean copulas, their density involves the generator derivatives,given by

(−1)dψ

(d)(t) = Pop(t1/β )/td , d ∈N, where Pop(x) =d

∑k=1

aGdk(1/β )(−1)k

ψ(k)(x)xk.

For an extension of this result, see [Hofert and Pham, 2013]. In this reference, densities ofnested Archimedean copulas are derived. They again heavily involve generator derivatives.

– As a special case of the construction of [Hofert and Vrins, 2013], Archimedean Sibuyacopulas arise. As a tractable subclass, asymmetric extensions of Archimedean copulas of theform

C(uuu) =ψV(∑

dj=1 ψ

−1µ j+V (u j)

)∏

dj=1 ψV (ψ

−1µ j+V (u j))

Π(uuu),

are presented, for a Laplace-Stieltjes transform ψV of a positive random variable V andcorresponding Laplace-Stieltjes transforms ψµ j+V of the random variables µ j +V for deter-ministic µ1, . . . ,µd . Their densities are given by

c(uuu) =d

∏j=1

1ψV (ψ

−1µ j+V (u j))

∑J⊆{1,...,d}

ψ(|J|)V

( d

∑j=1

ψ−1µ j+V (u j)

)

·∏j∈J

u j

ψ ′µ j+V (ψ

−1µ j+V (u j))

∏j/∈J

(1−

u jψ′V (ψ

−1µ j+V (u j))

ψ ′µ j+V (ψ

−1µ j+V (u j))ψV (ψ

−1µ j+V (u j))

),

thus again involve generator derivatives.High-order generator derivatives also appear in several statistically important quantities such as– the distribution function of the radial part

FR(t) = 1−d−1

∑k=0

(−1)kψ(k)(t)k!

tk, t ∈ [0,∞),

(here: a slightly simplified version), see [McNeil and Nešlehová, 2009];– the Kendall distribution function

K(t) =d−1

∑k=0

(−ψ−1(t))k

ψ(k)(ψ−1(t))/k!, t ∈ [0,1],

(again a slightly simplified version here), see [Barbe et al., 1996] or [Genest et al., 2011].




For statistical applications, it is therefore crucial to be able to accurately compute (or estimate)densities and generator derivatives of Archimedean copulas. A considerable amount of timehas gone into numerically stable and efficient implementation in the R package copula. Thesimulation results reported in this paper also serve as checks on the implementations in thepackage. Moreover, the numerical tricks and results presented also apply in the other contextsmentioned above.

We believe that issues of this type will become more important in the future as copula modelsin higher dimensions gain in applied and theoretical interest. Our computations will also point outinteresting (and partly surprising) results, which might lead to further research in this direction.

1.2. Overview and organization of the paper

There are several known approaches for estimating bivariate parametric Archimedean copulafamilies. Assuming the copula density to exist, maximum-likelihood estimation is one option;see [Genest et al., 1995] or [Tsukahara, 2005]. Another estimator resembles the method-of-moments estimator and consists of choosing the copula parameter such that a certain dependencemeasure, for example, Kendall’s tau, equals its empirical counterpart; see [Genest and Rivest,1993]. Although there is no theoretical justification for applying this method in more than twodimensions, using the mean of pairwise empirical Kendall’s taus and estimating the copulaparameter such that the population version of Kendall’s tau equals this mean also appears in theliterature; see [Berg, 2009] or [Savu and Trede, 2010]. A similar but different estimator is appliedin [Kojadinovic and Yan, 2010]. Another method in higher dimensions based on the moments ofthe Kendall distribution function is given in [Brahimi and Necir, 2011]. Other estimation methodsinclude approximating the probability integral transform with splines and using a minimumdistance approach between this distribution function and an empirical counterpart; see [Dimitrovaet al., 2008]. Splines also appear in [Lambert, 2007] for approximating a certain ratio involvingthe generator of the Archimedean copula to be estimated. [Tsukahara, 2005] considers minimumdistance estimators based on Cramér-von Mises or Kolmogorov-Smirnov distances and comparestheir performance to rank approximate Z-estimators in a simulation study involving the bivariateArchimedean Clayton, Frank, and Gumbel copula. Another estimation procedure in the bivariatecase is given by [Qu et al., 2010] based on minimizing a Cramér-von Mises distance between theempirical distribution function of a certain univariate random sample and the standard uniformdistribution. The approach described in [Stephenson, 2009] in the context of extreme-valuedistributions can be applied for estimating the parameter of a Gumbel copula in a Bayesiansetup. A non-parametric estimation procedure is introduced in [Genest et al., 2011]. For moregeneral information concerning copula parameter or copula density estimation in parametric and(especially) non-parametric set-ups, see [Charpentier et al., 2007].

In this work, we compare several known and new parametric estimators for Archimedeancopulas both under known and unknown margins (the margins being estimated non-parametrically).In the large-scale simulation study carried out, we compare the following estimators based onwell-known one-parameter generators (for two-parameter families, see [Hofert et al., 2012]):

1. We consider the method-of-moments estimator based on averaged pairwise sample versionsof Kendall’s tau. We also consider the average of pairwise Kendall’s tau estimators.




2. We apply a multivariate version of the measure of concordance known as Blomqvist’s betafor estimating Archimedean copulas. Blomqvist’s beta has the advantage of being givenexplicitly in terms of the copula. Similar to the method-of-moments estimation procedureintroduced by [Genest and Rivest, 1993], the copula parameters are estimated such that thepopulation and sample version of Blomqvist’s beta coincide.

3. We present several minimum distance estimators for estimating Archimedean copulas.Recently, a transformation of random variables following an Archimedean copula to uniformrandom variables (similar to Rosenblatt’s transformation but simpler to compute) wasintroduced by [Hering and Hofert, 2013]. The minimum distance estimators presented hereestimate the parameters as the minimum of certain Cramér-von Mises or Kolmogorov-Smirnov distances based on the transformation of [Hering and Hofert, 2013].

4. We consider maximum-likelihood estimation. Although the density of an Archimedeancopula has an explicit form in theory, deriving and evaluating the required derivatives isknown to be challenging from both a theoretical and a numerical perspective, especially inlarge dimensions; see also our motivation in Section 1.1. As mentioned below, computationsbased on computer algebra systems often fail already in low dimensions or require highprecision (and are therefore too slow to be applied, for example, in large-scale simulationstudies). We present explicit formulas for the densities of well-known Archimedean familiesand efficiently evaluate them. These results are based on the recent findings of [Hofert et al.,2012].

5. We introduce a simulated maximum-likelihood estimator to estimate Archimedean copulas.This estimator can be applied if the generator derivatives cannot be evaluated accuratelybut the copula is easy to sample.

6. We present maximum-likelihood estimation based on the diagonal of the Archimedeancopula. The main advantage is that the resulting estimation method is comparably easy andfast to apply in virtually any dimension.

The paper is organized as follows. In Section 2, we briefly recall the notion of Archimedeancopula. Section 3 introduces and presents the different estimators investigated in this work.Section 4 contains the large-scale simulation carried out. Section 5 addresses numerical issueswhen working in large dimensions and provides solutions to some of the problems mentioned.Section 6 concludes.

2. Archimedean copulas

Definition 2.1An (Archimedean) generator is a continuous, decreasing function ψ : [0,∞]→ [0,1] which satisfiesψ(0) = 1, ψ(∞) = limt→∞ ψ(t) = 0, and which is strictly decreasing on [0, inf{t : ψ(t) = 0}]. Ad-dimensional copula C is called Archimedean if it permits the representation

C(uuu) = ψ(t(uuu)), where t(uuu) =d

∑j=1

ψ−1(u j), uuu ∈ [0,1]d , (2)

for some generator ψ with inverse ψ−1 : [0,1]→ [0,∞], where ψ−1(0) = inf{t : ψ(t) = 0}.




[McNeil and Nešlehová, 2009] show that a generator defines an Archimedean copula of anydimension k ∈ {2, . . . ,d} if and only if ψ is d-monotone, that is, ψ is continuous on [0,∞], admitsderivatives up to the order d−2 satisfying (−1)k dk

dtk ψ(t)≥ 0 for all k ∈ {0, . . . ,d−2}, t ∈ (0,∞),and (−1)d−2 dd−2

dtd−2 ψ(t) is decreasing and convex on (0,∞). We mainly assume ψ to be completelymonotone, meaning that ψ is continuous on [0,∞] and (−1)k dk

dtk ψ(t)≥ 0 for all k ∈N0, t ∈ (0,∞),so that ψ is the Laplace-Stieltjes transform L S [F ] of a distribution function F on the positivereal line; see Bernstein’s Theorem in [Feller, 1971, p. 439]. The class of all such generators isdenoted by Ψ∞ and it is clear that a ψ ∈Ψ∞ generates an Archimedean copula in any dimensionsd.

There are several known parametric Archimedean generators (see, for example, [Nelsen, 2006,pp. 116]) also referred to as Archimedean families. Among the most widely used in applications arethose of Ali-Mikhail-Haq (A), Clayton (C), Frank (F), Gumbel (G), and Joe (J). We will considerthese generators as working examples; see Table 1 which also includes population versions ofKendall’s tau for these families. Here, D1(θ) =

∫θ

0 t/(exp(t)−1)dt/θ denotes the Debye functionof order one. Detailed information about the distribution functions F corresponding to the givengenerators can be found in [Hofert, 2012] and references therein.

TABLE 1. Well-known one-parameter Archimedean generators ψ with corresponding Kendall’s tau. The range ofattainable Kendall’s tau is (0,1/3) for A, (0,1) for C and F, and [0,1) for G and J.

Family Parameter ψ(t) τ

A θ ∈ [0,1) (1−θ)/(exp(t)−θ) 1−2(θ +(1−θ)2 log(1−θ))/(3θ 2)

C θ ∈ (0,∞) (1+ t)−1/θ θ/(θ +2)F θ ∈ (0,∞) − log

(1− (1− e−θ )exp(−t)

)/θ 1+4(D1(θ)−1)/θ

G θ ∈ [1,∞) exp(−t1/θ ) (θ −1)/θ

J θ ∈ [1,∞) 1− (1− exp(−t))1/θ 1−4∑∞k=1 1/(k(θk+2)(θ(k−1)+2))

3. Estimation methods for Archimedean copulas

Assume that we have given realizations xxxi, i∈{1, . . . ,n}, of independent and identically distributed(iid) random vectors XXX i, i∈ {1, . . . ,n}, from a joint distribution function H with known margins Fj,j ∈ {1, . . . ,d}, Archimedean copula C generated by ψ , and corresponding density c. The generatorψ is assumed to belong to a parametric family (ψ

θθθ)θθθ∈Θ with parameter vector θθθ ∈ Θ ⊆ Rp,

p ∈N, and the true but unknown vector is θθθ 0 (similarly, C =Cθθθ 0 and c = cθθθ 0). If the marginsFj, j ∈ {1, . . . ,d}, are known, ui j = Fj(xi j), i ∈ {1, . . . ,n}, j ∈ {1, . . . ,d}, is a random samplefrom C. In practice, the margins are typically unknown and must be estimated parameterically ornon-parametrically. In the following, whenever working under unknown margins, we will assumethe latter approach and thus consider the pseudo-observations

ui j =n

n+1Fn, j(xi j) =

ri j

n+1, (3)

where Fn, j denotes the empirical distribution function corresponding to the jth margin and ri j

denotes the rank of xi j among all xi j, i ∈ {1, . . . ,n}.For estimating θθθ 0, we now present several methods, some of which (such as the simulated

or the diagonal maximum-likelihood estimator) are new. We give the formulas in terms of a




random sample UUU i, i ∈ {1, . . . ,n}, from C. In Section 4, this random sample is replaced either byrealizations uuui, i∈ {1, . . . ,n} (when working under known margins) or by the pseudo-observationsuuui, i ∈ {1, . . . ,n}, when working under unknown margins.

3.1. Pairwise Kendall’s tau

Kendall’s tau is defined to be

τ =E[sign((X1−X ′1)(X2−X ′2))],

where (X1,X2)> is a vector of two continuously distributed random variables, (X ′1,X

′2)> is an

independent copy of (X1,X2)>, and sign(x) = 1(0,∞)(x)−1(−∞,0)(x) denotes the signum function.

Kendall’s tau is a measure of concordance (see [Scarsini, 1984]) and therefore measures thestrength of association (as a number in [−1,1]) between large values of one variable and largevalues of the other. Note that Archimedean copulas with generator ψ ∈Ψ∞ are positive lowerorthant dependent, thus Kendall’s tau always lies in [0,1] for such copulas; see, for example,[Hofert, 2010, pp. 59]. Kendall’s tau has an obvious estimator, referred to as the sample version ofKendall’s tau. Based on the random sample UUU i = (Ui1,Ui2)

>, i ∈ {1, . . . ,n}, it is given by

τn =

(n2

)−1

∑1≤i1<i2≤n

sign((Ui11−Ui21)(Ui12−Ui22)).

It can also be estimated directly from the bivariate sample XXX i, i ∈ {1, . . . ,n}.If C is a bivariate Archimedean copula generated by a twice continuously differentiable

generator ψ with ψ(t)> 0 for all t ∈ [0,∞), Kendall’s tau can be represented in semi-closed formas

τ = 1+4∫ 1

0

ψ−1(t)(ψ−1(t))′

dt = 1−4∫

∞

0t(ψ ′(t))2 dt

(see [Joe, 1997, p. 91]) which can often be computed explicitly; see Table 1.[Genest and Rivest, 1993] introduce a method-of-moments estimator for bivariate one-parameter

Archimedean copulas based on Kendall’s tau. The copula parameter θ0 ∈Θ⊆R is estimated byθn such that

τ(θn) = τn,

where τ(θ) denotes Kendall’s tau of the corresponding Archimedean family viewed as a functionof the parameter θ ∈Θ⊆R. In other words,

θn = τ−1(τn), (4)

assuming the inverse τ−1 of τ exists. This estimation method obviously only applies to one-parameter families. Otherwise, the set of all parameters with equal Kendall’s tau is a level curveand so Kendall’s tau cannot be uniquely inverted. If (4) has no solution, this estimation methoddoes not lead to an estimator. Note that unless there is an explicit form for τ−1, θn is computed bynumerical root finding.




[Berg, 2009] and [Savu and Trede, 2010] apply this method to data of dimension d > 2 byusing pairwise sample versions of Kendall’s tau. If τn, j1 j2 denotes the sample version of Kendall’stau between the j1th and j2th data column, then θ is estimated by

θn = τ−1

((d2

)−1

∑1≤ j1< j2≤d

τn, j1 j2

). (5)

We denote this estimator or estimation method by τ ¯τ . Intuitively, the parameter is chosen suchthat Kendall’s tau equals the average over all pairwise sample versions of Kendall’s tau. Notethat properties of this estimator are not known and also not easy to derive since the average istaken over dependent data columns. In particular, although

(d2

)−1∑1≤ j1< j2≤d τn, j1 j2 is unbiased

for τ(θ0), the estimator in (5) need not be unbiased for θ0.Another “pairwise” estimator can be obtained by first computing the

(d2

)pairwise estimators as

given in (4) and then average over the estimators, that is,

θn =

(d2

)−1

∑1≤ j1< j2≤d

τ−1(τn, j1 j2).

This unbiased estimator can be found in [Kojadinovic and Yan, 2010]; see, for example, thefunction fitCopula(, method=“itau”) in the R package copula. We denote it or the corre-sponding estimation method by τ ¯

θ.

3.2. Blomqvist’s beta

Blomqvist’s beta (see, for example, [Nelsen, 2006, p. 182]) is also a measure of concordance. Inthe bivariate case with X j ∼ Fj, j ∈ {1,2}, it is defined by

β =P((X1−F−1 (1/2))(X2−F−2 (1/2))> 0)−P((X1−F−1 (1/2))(X2−F−2 (1/2))< 0)

and therefore measures the probability of falling into the first or third quadrant minus the proba-bility of falling into the second or fourth quadrant, the quadrants being defined by the mediansF−j (1/2), j ∈ {1,2}. This measure can be expressed in terms of the copula of (X1,X2)

>. It alsoallows for a natural generalization to d > 2, given by

β =2d−1

2d−1−1(C(1/2, . . . ,1/2)+C(1/2, . . . ,1/2)−21−d);

see, for example, [Schmid and Schmidt, 2007]. Here, C denotes the survival copula correspondingto C. For Archimedean copulas as given in (2), Blomqvist’s beta is easily seen to be

β =2d−1

2d−1−1

(ψ(dψ

−1(1/2))+( d

∑j=0

(dj

)(−1) j

ψ( jψ−1(1/2)))−21−d

). (6)

Given the random sample UUU i, i ∈ {1, . . . ,n}, the sample version of Blomqvist’s beta is given by

βn =2d−1

2d−1−1

(1n

n

∑i=1

( d

∏j=11{Ui j≤1/2}+

d

∏j=11{Ui j>1/2}

)−21−d

)(7)




For asymptotic properties of βn, see [Schmid and Schmidt, 2007].

A method-of-moments estimator based on Blomqvist’s beta can be obtained via

θn = β−1(βn),

where β (θ) denotes β as a function of the parameter θ ∈ Θ ⊆ R. We denote this estimatoror estimation method by β . As for Kendall’s tau, this estimation method only applies to theone-parameter case. Typically, θn is computed via numerical root finding.

3.3. Minimum distance estimation

[Hering and Hofert, 2013] present a transformation for Archimedean copulas that is analogousto Rosenblatt’s transform but simpler to compute. Consider a d-monotone generator ψ and letUUU follow the Archimedean copula C with generator ψ . Furthermore, let the Kendall distributionfunction K (that is, the distribution function of the probability integral transformation C(UUU)) becontinuous. Then, the transformed random vector UUU ′′′ = Tψ(UUU) with

U ′j =

(∑

jk=1 ψ−1(Uk)

∑j+1k=1 ψ−1(Uk)

) j

, j ∈ {1, . . . ,d−1}, U ′d = K(C(UUU)) (8)

follows a uniform distribution on [0,1]d , denoted by UUU ′′′ ∼ U[0,1]d . Note that if ψ ∈ Ψ∞, then

K(t) = ∑d−1k=0

ψ(k)(ψ−1(t))k! (−ψ−1(t))k; see [Barbe et al., 1996] or [McNeil and Nešlehová, 2009].

The transformation (8) allows one to easily derive a minimum distance estimator. First, onetransforms the random vectors UUU i, i∈{1, . . . ,n}, with Tψ and then minimizes a “distance” betweenthe transformed variates and the multivariate uniform distribution. This could be achieved, forexample, with the statistics S(B)n or S(C)

n used by [Genest et al., 2009]. For simplicity and run-timeperformance, however, we map the transformed variates to univariate quantities via

Y ni =

d

∑j=1

(Φ−1(U ′i j))2 or Y l

i =d

∑j=1− logU ′i j, i ∈ {1, . . . ,n},

where Φ−1 denotes the quantile function of the standard normal distribution. Such mappingsto a univariate setting are known from goodness-of-fit testing; see [D’Agostino and Stephens,1986, p. 97]. If the transformation Tψ(UUU) is applied with the correct parameter, then Y n

i ∼ Fχ2

dand

Y li ∼ FΓd , i ∈ {1, . . . ,n}, that is, Y n

i and Y li should follow a chi-square distribution with d degrees

of freedom and a Γ(d,1) distribution, respectively. Hence, minimum distance estimators can be




obtained via the Cramér-von Mises and Kolmogorov-Smirnov type of distances

θθθn,CvMn = arginf

θθθ∈Θ

n∫

∞

−∞

∣∣Fn,Y n(x)−Fχ2

d(x)∣∣2 dF

χ2d(x)

= arginfθθθ∈Θ

112n

+n

∑i=1

(2i−1

2n−F

χ2d(Y n

(i))

)2

,

θθθn,KSn = arginf

θθθ∈Θ

supx

∣∣Fn,Y n(x)−Fχ2

d(x)∣∣

= arginfθθθ∈Θ

maxi∈{1,...,n}

{F

χ2d(Y n

(i))−i−1

n,

in−F

χ2d(Y n

(i))

},

θθθl,CvMn = arginf

θθθ∈Θ

n∫

∞

−∞

∣∣Fn,Y l(x)−FΓd (x)∣∣2 dFΓd (x)

= arginfθθθ∈Θ

112n

+n

∑i=1

(2i−1

2n−FΓd (Y

l(i))

)2

,

θθθl,KSn = arginf

θθθ∈Θ

supx

∣∣Fn,Y l(x)−FΓd (x)∣∣

= arginfθθθ∈Θ

maxi∈{1,...,n}

{FΓd (Y

l(i))−

i−1n

,in−FΓd (Y

l(i))

},

where Fn,Y n and Fn,Y l denote the empirical distribution functions based on (Y ni )i∈{1,...,n} and

(Y li )i∈{1,...,n}, respectively, and Y n

(i) and Y l(i), i∈{1, . . . ,n}, denote the order statistics of (Y n

i )i∈{1,...,n}

and (Y li )i∈{1,...,n}, respectively. We denote these four estimators or estimation methods by MDECvM

χ ,MDEKS

χ , MDECvMΓ

, and MDEKSΓ

, respectively.In large dimensions, one can omit the possibly costly computation of U ′d and work with U ′j,

j ∈ {1, . . . ,d− 1}, only, see [Hering, 2011, pp. 52]. Note that minimum distance estimatorsnaturally also work for p≥ 2, that is, parameter vectors θθθ ∈Θ⊆Rp.

3.4. Maximum-likelihood estimation

According to [McNeil and Nešlehová, 2009], an Archimedean copula C admits a density c if andonly if ψ(d−1) exists and is absolutely continuous on (0,∞). In this case, c is given by

c(uuu) = ψ(d)(t(uuu))

d

∏j=1

(ψ−1)′(u j) =ψ(d)(t(uuu))

∏dj=1 ψ ′(ψ−1(u j))

, uuu ∈ (0,1)d , (9)

where, as in (2), t(uuu) = ∑dj=1 ψ−1(u j). Note that for computing the log-density, it is convenient to

write c as

c(uuu) = (−1)dψ

(d)(t(uuu))d

∏j=1−(ψ−1)′(u j) =

(−1)dψ(d)(t(uuu))

∏dj=1−ψ ′(ψ−1(u j))

.




Given the sample UUU i, i ∈ {1, . . . ,n}, finding the maximum-likelihood estimator (MLE) usuallyinvolves solving the optimization problem

θθθ n = argsupθθθ∈Θ

n

∑i=1

logcθθθ (UUU i),

where here and in the following the subscript θθθ is used to stress the dependence on θθθ . Thisrequires an efficient strategy for evaluating the (log-)density. The most important part is to knowhow to derive and compute the generator derivatives. Tools like automatic differentiation, see[Griewank and Walther, 2003], might provide a solution. Recently, [Hofert et al., 2012] presentedexplicit formulas for all families listed in Table 1. The corresponding copula densities are reportedhere for the reader’s convenience (note that α = 1/θ ):

1. For the family of Ali-Mikhail-Haq,

cθ (uuu) =(1−θ)d+1

θ 2hA

θ(uuu)

∏dj=1 u2

jLi−d(hA

θ (uuu)),

where Li−s(z) = ∑∞k=1

zk

ks is the polylogarithm of order s at z and hAθ(uuu) = θ ∏

dj=1

u j1−θ(1−u j)

.

2. For the family of Clayton,

cθ (uuu) =d−1

∏k=0

(θk+1)( d

∏j=1

u j

)−(1+θ)

(1+ tθ (uuu))−(d+α).

3. For the family of Frank,

cθ (uuu) =(

θ

1− e−θ

)d−1

Li−(d−1)(hFθ (uuu))

exp(−θ ∑dj=1 u j)

hFθ(uuu)

,

where hFθ(uuu) = (1− e−θ )1−d

∏dj=1(1− exp(−θu j)).

4. For the family of Gumbel,

cθ (uuu) = θd exp(−tθ (uuu)α)

∏dj=1(− logu j)

θ−1

tθ (uuu)d ∏dj=1 u j

PGd,α(tθ (uuu)

α),

where

PGd,α(x) =

d

∑k=1

aGdk(α)xk,

aGdk(α) = (−1)d−k

d

∑j=k

αjs(d, j)S( j,k) =

d!k!

k

∑j=1

(kj

)(α jd

)(−1)d− j, k ∈ {1, . . . ,d},

and s and S denote the Stirling numbers of the first kind and the second kind, respectively,given by the recurrence relations

s(n+1,k) = s(n,k−1)−ns(n,k),

S(n+1,k) = S(n,k−1)+ kS(n,k),

for all k∈N, n∈N0, with s(0,0) = S(0,0) = 1 and s(n,0) = s(0,n) = S(n,0) = S(0,n) = 0for all n ∈N.




5. For the family of Joe,

cθ (uuu) = θd−1 ∏

dj=1(1−u j)

θ−1

(1−hJθ(uuu))1−1/θ

PJd,α

(hJ

θ(uuu)

1−hJθ(uuu)

),

where

PJd,α(x) =

d−1

∑k=0

aJdk(α)xk,

aJdk(α) = S(d,k+1)(k−α)k, k ∈ {0, . . . ,d−1},

hJθ(uuu) = ∏

dj=1(1− (1−u j)

θ ), and (k−α)k =Γ(k+1−α)

Γ(1−α) denotes the falling factorial.

Example 3.1The left-hand side of Figure 1 shows the log-likelihood of a Clayton copula based on a 100-dimensional sample of size n = 100 with parameter θ0 = 2 such that the corresponding bivariatepopulation version of Kendall’s tau equals τ(θ0) = 0.5. The MLE is denoted by θn. The right-handside of Figure 1 shows the log-likelihood plot of a 100-dimensional Gumbel family with parameterθ0 = 2 such that Kendall’s tau equals τ(θ0) = 0.5. Both Figures are plotted on the interval[τ−1(τ(θ0)−h),τ−1(τ(θ0)+h)], where h = 0.05 denotes a “distance” in terms of concordance.Note that evaluating the log-density of a Gumbel copula is numerically highly complicated; seeSection 5.3 for more details.

θ

n = 100 d = 100 θ0 = 2 τ(θ0) = 0.5

l(θ; u

1,…

, un)

log−likelihood of a Clayton copula

6800

6850

6900

6950

1.7 1.8 1.9 θ0θn 2.1 2.2 2.3 2.4

θ

n = 100 d = 100 θ0 = 2 τ(θ0) = 0.5

l(θ; u

1,…

, un)

log−likelihood of a Gumbel copula

5680

5700

5720

5740

5760

1.85 1.90 1.95 θ0 θn 2.05 2.10 2.15 2.20

FIGURE 1. Plot of the log-likelihood of a Clayton (left) and Gumbel (right) copula based on a sample of size n = 100in dimension d = 100 with parameter θ0 = 2 such that Kendall’s tau equals 0.5.




3.5. Simulated maximum-likelihood estimation

If the derivatives of a given Archimedean generator ψ are not known explicitly one may usethe fact that ψ is an expectation in order to approximate the density of the generated copula.This way one can replace derivatives of higher order by just one integral (which can either beevaluated numerically or via Monte Carlo simulation). If ψ ∈ Ψ∞, then ψ(t) = L S [F ](t) =∫

∞

0 exp(−xt)dF(x), so that by differentiating under the integral sign one obtains

(−1)dψ

(d)(t) =∫

∞

0xd exp(−xt)dF(x) =E[V d exp(−Vt)], t ∈ (0,∞),

where V has distribution function F . An approximation to (−1)dψ(d)(t) is thus given by

(−1)dψ

(d)(t)≈ 1m

m

∑k=1

V dk exp(−Vkt), t ∈ (0,∞), (10)

where Vk ∼ F , k ∈ {1, . . . ,m}, are realizations of iid random variables following F = L S −1[ψ].Instructions for how to sample F for the one-parameter families in Table 1 can be found, forexample, in [Hofert, 2012]; see also [Hofert and Mächler, 2011]. This method can be used toevaluate the copula density. We refer to the corresponding MLE as simulated maximum-likelihoodestimator (SMLE). Finally, note that both the MLE and the SMLE naturally apply to the multi-parameter case.

3.6. Diagonal maximum-likelihood estimation

The diagonal δ (u) =C(u, . . . ,u) of a copula C suggests a simple and straightforward estimationprocedure which, as far as we are aware, has not been exploited for estimating (Archimedean)copulas so far. Note that the diagonal δθθθ of a parametric copula family (Cθθθ )θθθ∈Θ is a distributionfunction and, for UUU ∼C,

Y = max1≤ j≤d

U j ∼ δθθθ .

Based on the sample UUU i, i ∈ {1, . . . ,n}, with corresponding maxima Yi, i ∈ {1, . . . ,n}, one canapply maximum-likelihood estimation to find an estimator θθθ n of the parameter vector θθθ 0 via

θθθ n = argsupθθθ∈Θ

n

∑i=1

logδ′θθθ(Yi), (11)

where δ ′θθθ

denotes the density of the distribution function δθθθ . We refer to the estimator θθθ n as diag-onal maximum-likelihood estimator (DMLE). For Archimedean copulas, δθθθ (u) = ψθθθ (dψ

−1θθθ

(u))and hence,

δ′θθθ(u) = dψ

′θθθ(dψ

−1θθθ

(u))(ψ−1θθθ

)′(u), u ∈ [0,1]. (12)

Therefore, one advantage of the DMLE is that the degree of numerical difficulty of the optimizationin (11) (theoretically) remains rather unaffected by the dimension. Another advantage may be its




computational speed, which makes it useful for computing initial values for more sophisticatedestimators. As a drawback, the DMLE only uses information from the copula diagonal.

For the one-parameter Gumbel family, (11) can be solved explicitly, the estimator θ Gn of θ

being

θGn =

logdlogn− log

(∑

ni=1− logYi

) .An adjusted estimator of the form

θG,∗n = max{θ G

n ,1}

is then guaranteed to provide an admissible parameter estimator for Gumbel’s family.

4. A large-scale simulation study

In this section, we present a large-scale simulation study in which we compare the performanceof the different estimators presented in Section 3 both under known and (from an applied point ofview the more interesting case of) unknown margins (pseudo-observations). To the best of ourknowledge, this is the first study of this kind also addressing large dimensions (up to 100). To beable to also include the estimators based on measures of concordance, we restrict ourselves to theone-parameter families as given in Table 1.

4.1. A word concerning the implementation

The results presented in this section are based on the following computational set-up. Theprocedures are computationally challenging in many ways and much effort has gone into accurateand efficient implementation in R; see Section 5. The latest version of the package can be accessedvia http://nacopula.r-forge.r-project.org/. The computations are carried out on thecomputer cluster Brutus of ETH Zurich which runs CentOS 5.4. The batch jobs are run on nodeswith four quad-core AMD Opteron 8380 CPUs and 32 GB of RAM. Apart from the physicalstructure of the grid, the compiler, and the programming language, note that run time also dependson other factors such as the quality of the implementation or the current load of the machine. Thepresented run times should therefore be viewed with this in mind.

4.2. The experimental design

In the simulation study carried out, we consider both known and unknown margins. For eachof these cases, we generate N = 1000 samples of size n from iid random vectors following ad-dimensional Archimedean copula with prespecified parameter θ such that the correspondingKendall’s tau is τ ∈ {0.25,0.75}. For the case of known margins, we base estimation directlyon these samples. For the case of unknown margins, we base estimation on the correspondingpseudo-observations as given in (3). Since we are mainly interested in the behavior for differentdimensions d, we consider d ∈ {5,20,100} and keep n = 100 fixed. The dimensions consideredthus go up to the sample size n in which case the data matrices have 10000 entries; note that such



http://nacopula.r-forge.r-project.org/


a setup is computationally highly challenging for all the estimators we consider and a novelty inthe context of dependence modeling with copulas. We investigate the one-parameter families ofAli-Mikhail-Haq (only for τ = 0.25 since the range of admissible Kendall’s tau for this family isbounded from above by 1/3), Clayton, Frank, Gumbel, and Joe. The average pairwise Kendall’stau estimator (τ ¯τ ), the average of Kendall’s tau estimators (τ ¯

θ), the estimation method based

on Blomqvist’s beta (β ), the four presented minimum distance estimators (MDECvMχ , MDEKS

χ ,MDECvM

Γ, and MDEKS

Γ), maximum-likelihood (MLE), simulated maximum-likelihood (SMLE),

and diagonal maximum-likelihood (DMLE) estimators are applied to estimate the parameter foreach of the N data sets. Finally, bias and root mean squared error (RMSE), as well as mean userrun time over all N replications are computed.

For the required optimizations for the estimators based on Blomqvist’s beta, all minimumdistance estimators, MLE, SMLE, and DMLE, we use initial intervals determined from a largerange of (admissible) Kendall’s tau; see the implementation of the function initOpt in theR package copula for more details. For the minimum distance estimators to be competitiveaccording to run time, we only include the Kendall distribution function in the transformationgiven in (8) in the five-dimensional case, but not for higher dimensions. For applying the SMLE,we draw 10000 random variates from V ∼ F = L S [ψ] for each evaluation of the density of theArchimedean copula.

4.3. Results under known margins

Tables 2, 3, and 4 in the appendix contain the bias (multiplied by 1000), the RMSE (multiplied by1000), and the mean user times in milliseconds (MUT), respectively, for all investigated estimatorsunder known margins. For each entry, the number in parentheses denotes the entry divided by thecorresponding entry of the MLE column, so that the performance with respect to the MLE caneasily be determined; the MLE itself thus has always 1.0 in parentheses. For the RMSEs, note thatthe reciprocals of the square of these numbers are also known as the (estimated) relative efficiencyof the MLE with respect to the corresponding estimator.

Figures 2 and 3 graphically display the square root of the absolute error via box plots obtainedfrom the N = 1000 replications. Due to readability, we exclude methods which perform so poorlythat their boxplots dominate the scale of values. In particular, this is the case for the estimatorsbased on Blomqvist’s beta for d = 100 for the families of Clayton, Frank, and Joe and the SMLEfor Clayton and τ = 0.25.

The results from this study under known margins can be summarized as follows:– The performance of the average of bivariate Kendall’s tau estimator τ ¯

θas given in the end

of Section 3.1 is very similar to the one of the averaged pairwise Kendall tau estimator τ ¯τ .One problem that especially τ ¯

θfaces is that sample versions of Kendall’s tau are sometimes

not in the range of tau as a function of theta. These values were then mapped to the rangeof admissible Kendall’s tau, see the function tau.checker in copula. Furthermore, runtime for method τ ¯

θis typically larger (especially in large dimensions) than that for τ ¯τ , which

is clear since more inversions of Kendall’s tau have to be performed. Overall, τ ¯τ is thuspreferred. Furthermore, due to only considering pairs at a time, this estimator is quite robustagainst numerical difficulties. A disadvantage, however, is its large run time due to thequadratic complexity in the dimension d.




– Although Blomqvist’s beta can be flawlessly applied to estimate the copula parameter θ

for small and moderate dimensions d, this estimator shows serious numerical problems forlarge values of d uniformly over all investigated Archimedean families. One of the problemsturns out to be that both products appearing in the sample version (7) of Blomqvist’s betaare sometimes zero, so that βn < 0 although β ≥ 0. Another problem is that the evaluationof the survival copula at (1/2, . . . ,1/2)> turns out to be numerically challenging for severalfamilies; see Section 5.6 for more details.

– The performance of the minimum distance estimators depends on the mapping to the one-dimensional setting applied. In particular, the estimators MDECvM

Γand MDEKS

Γbased on

the logarithmic transformation to a Gamma distribution do not perform well in comparisonto MDECvM

χ and MDEKSχ . Furthermore, the minimum distance estimators based on the

Kolmogorov-Smirnov distances are outperformed by those based on the Cramér-von Misesdistances (also according to run time in most of the cases investigated, see Table 4). Note thatrun time for d = 5 is larger than for d ∈ {20,100} because we applied the full transformationTψ including the Kendall distribution function K in the five-dimensional case. Moreover, thedistances (objective functions) had to be reparameterized in order for the minimum distanceestimators to be computed; see Section 5.1 for more details.

– With the explicit formulas for the densities we presented, the MLE clearly shows the bestperformance under known margins. Note that the run times are much smaller than onewould expect in comparison to other estimators, although, our implementation was writtenwith focus on readability rather than run-time performance and thus several quantities arecomputed each time the density is evaluated. In contrast to statements found in the literature(see, for example, [Berg and Aas, 2009] or [Weiß, 2010]) this leaves no doubt that maximum-likelihood estimation is feasible in large dimensions and performs well; for the latter, seealso [Hofert et al., 2012] who empirically show that the mean squared error (MSE) satisfies

MSE ∝1

nd

in the case of known margins.– The SMLE also shows an incredible performance, the only exception being Clayton’s family;

see Section 5.2 for more details. The RMSEs are close to the ones obtained by maximum-likelihood estimation. Moreover, this method is straightforward to implement given randomnumber generators for the distribution corresponding to the generator under consideration.The drawback of this method is certainly a larger run time. Note that this could be partlyreduced, for example, by using an adaptive technique in which smaller amounts of randomvariates are drawn during the first couple of steps the optimizer performs. Also note thatwith the constant use of 5000 random variates (instead of 10 000) per density evaluation, theoverall performance of the SMLE is still slightly better than those of the minimum distanceestimator MDECvM

χ .– The advantage of the DMLE lies in its speed. Due to this fact, this estimator could be used

for finding initial values for more sophisticated estimation methods.




Estimator

θ n−

θ,

θ=

τ−1(0

.25)

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

1.0

1.2

0.5

1.0

1.5

2.0

0.2

0.4

0.6

0.8

1.0

1.2

0.5

1.0

1.5

d = 5

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MD

EC

vMc

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EK

Sc

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EC

vMg

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EK

Sg

MLE

SM

LE

DM

LE

d = 20

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FIGURE 2. Box plots of the square root of the absolute error (under known margins) obtained from N = 1000replications of sample size n = 100 for Kendall’s tau equal to 0.25.




Estimator

θ n−

θ,

θ=

τ−1(0

.75)

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EC

vMc

MD

EK

Sc

MD

EC

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MD

EK

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SM

LE

DM

LE

Clayton

Frank

Gum

belJoe

FIGURE 3. Box plots of the square root of the absolute error (under known margins) obtained from N = 1000replications of sample size n = 100 for Kendall’s tau equal to 0.75.




4.4. Results under unknown margins

Tables 5, 6, and 7 in the appendix contain the bias (multiplied by 1000), the RMSE (multiplied by1000), and the mean user times in milliseconds (MUT), respectively, for all investigated estimatorsbased on pseudo-observations.

Figures 4 and 5 graphically display the corresponding square root of the absolute error via boxplots obtained from the N = 1000 replications. As for Figures 2 and 3, we exclude methods whichperform so poorly that their boxplots dominate the scale of values. Under pseudo-observations,these were the same methods as under known margins with the only exceptions being MDEKS

Γ

for d ∈ {20,100} which are excluded and SMLE for Clayton which performed better underpseudo-observations and is thus included in the figures; see Section 5.2 for an explanation.

The performance of the estimators based on pseudo-observations can be summarized as follows.Overall, the MLE still performs best, but the differences in absolute error are much less obvious.Furthermore, although a slight improvement of the performance of the MLE in larger dimensionsd is visible, the rate of improvement does not seem to be as large as under known margins.Concerning the estimators based on Kendall’s tau and the minimum distance estimators, theformer performs well for the case τ = 0.25, the latter performs well for the case τ = 0.75.

5. Numerical issues and partial solutions

In this section, we address some specific numerical problems we encountered when working inhigh dimensions. These problems are not trivial to solve and for some, no simple solution existsto date. We included this section to emphasize that working in large dimensions is much moreaffected by numerical issues. This is not merely a problem of slow run times; it is also a hugeproblem for precision. As a general remark, let us stress that what is known about estimatorsin low dimensions does not always carry over to the high-dimensional case: Estimators that arefast in low dimensions may turn out to be too slow in large dimensions (although robust, thepairwise Kendall’s tau estimators face this problem); estimators whose simple form suggest goodperformance in large dimensions may be highly prone to numerical errors (which is the case, forexample, for Blomqvist’s beta due to accessing the survival copula involved, a critical task inlarge dimensions).

5.1. Minimum distance estimators

The minimum distance estimators were especially prone to the problem of a flat objective functionfor the optimization for all but the Ali-Mikhail-Haq family. Note that there, the parameter θ runsin a bounded interval which is typically advantageous for optimization.

To show the problem, we consider the Gumbel copula and pick out the Cramér-von Misesdistances based on the mapping to a χ2 distribution (via the quantile function of the normaldistribution) as described in Section 3.3. The left-hand side of Figure 6 shows the objectivefunction (the distance to be minimized) based on samples of size n = 100 from Gumbel copulas inthe dimensions d ∈ {5,20,100} with parameter θ = 4/3 such that Kendall’s tau equals 0.25. Wechoose the same (large) plotting interval as is chosen for the optimization in the simulation studyin Section 4. As can be seen from this figure, the distance to be minimized becomes flat already




Estimator

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θ,

θ=

τ−1(0

.25)

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laytonF

rankG

umbel

Joe

FIGURE 4. Box plots of the square root of the absolute error (under unknown margins) obtained from N = 1000replications of sample size n = 100 for Kendall’s tau equal to 0.25.




Estimator

θ n−

θ,

θ=

τ−1(0

.75)

0.5

1.0

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LE

Clayton

Frank

Gum

belJoe

FIGURE 5. Box plots of the square root of the absolute error (under unknown margins) obtained from N = 1000replications of sample size n = 100 for Kendall’s tau equal to 0.75.




for moderate parameter values. The optimization of this distance carried out in the simulationstudy is done via R’s optimize. It is indicated on the corresponding help page how this functionproceeds. Based on the first two points in the optimization procedure, it is clear that the algorithmremains in the “flat part” of the distance function and thus returns wrong estimates.

The solution to this problem is simple and effective: By reparameterizing the distance one cancarry out the optimization without problems. To see why, consider the right-hand side of Figure 6which shows precisely the same distance as on the left-hand side of this figure, but now plottedin α = 1−1/θ . The advantage of this reparameterization is that the objective function is now afunction of the bounded variable α ∈ (0,1].

05

1015

2025

30

θ

Gum

bel:

CvM

dis

tanc

e to

be

min

imiz

ed

1 5 10 15 20

d = 5d = 20d = 100

05

1015

2025

30

α

Gum

bel:

CvM

dis

tanc

e to

be

min

imiz

ed

0.0 0.2 0.4 0.6 0.8 1.0

d = 5d = 20d = 100

FIGURE 6. Plot of the Cramér-von Mises distances (based on the mapping to a χ2 distribution) without (left) and with(right) reparameterization of the distance for the Gumbel copula with parameter θ = 4/3 (Kendall’s tau equals 0.25)based on a sample of size n = 100 in the dimensions indicated.

Similar transformations turn out to be convenient for the families of Clayton, Frank, and Joe aswell. For the latter, we use the same reparameterization as for Gumbel, for Clayton and Frank weuse α = 2arctan(θ)/π .

5.2. Simulated maximum-likelihood estimation

As can be seen from the results in Sections 4.3 and 4.4, the SMLE performs well except forClayton’s family under known margins. In this section we briefly investigate why. For this, recallthat the SMLE is based on the approximation (10). The log-density approximated via this MonteCarlo method then involves

log((−1)dψ

(d)(t))≈ log(

1m

m

∑k=1

V dk exp(−Vkt)

)= log

(1m

m

∑k=1

exp(bk)

), (13)

where bk = d log(Vk)−Vkt.

For the SMLE to compute, we have to replace t in (13) by ∑dj=1 ψ−1(u j). For simplicity, let us

assume that all components u j are equal to u, so we consider the vector uuu = (u, . . . ,u)> ∈ [0,1]d .




The corresponding value t for (13) is then t = dψ−1(u) = d(u−θ −1). Let us assume that θ = 2,that is, the corresponding value of Kendall’s tau is 0.5. If u is small, then t becomes large. Theproblem is now that the exponents bk become quite small. Indeed, they are so small (dependingon t) that the exp(bk) become zero in computer arithmetic for many (again depending on t) of them = 10000 sampled Vk’s. These zeros significantly affect the approximation in (13).

To give an example, let θ = 2, d = 5, draw iid Vk ∼ Γ(1/θ ,1), k ∈ {1, . . . ,m}, for m = 10000(using set.seed(1)), and compute the approximation in (13) at t ∈ {5 ·1016,5 ·1012,5 ·108,15}.The left-hand side (correct values) for these values of t are (roughly) -208.09, -157.44, -106.78,and -11.86, whereas the right-hand side gives -Inf, -622.62, -124.41, and -11.86. As one can see,for t = 15 both values agree, for t = 5 · 108, the approximation is already quite far away fromthe corresponding true value. For large t this problem becomes more severe, with the extremecase being such a large t that exp(bk) is zero in computer arithmetic for all Vk’s. This impliesthat log(0) =−Inf in (13). Note that this could be avoided by using an intelligent logarithm asgiven in Lemma 5.1 1 below. However, this does not solve the problem of a poor approximationto log((−1)dψ(d)(t)) (see also Figure 7 below), the problem being that all summands are zero,except the one being exp(bk−bmax) = exp(0) = 1.

Figure 7 shows, in log-log scale, the relative error of the approximation (13) for uuu = (u, . . . ,u)>

as a function of u, based on m = 10000, d = 5, and θ = 2. As is clearly visible, the relative errorof the approximation becomes much larger for smaller values of u. Since the Clayton copulahas lower tail dependence, there is indeed a positive probability of obtaining random vectorswith simultaneously small components. These (and only these) samples affect the likelihoodapproximation and lead to wrong SMLEs. Note that this problem vanishes for the SMLE based onpseudo-observations, see Figures 4 and 5, since each ui j ∈ {1/(n+1), . . . ,n/(n+1)} and thus theui j’s are naturally bounded from below by 1/(n+1). Another option (not discussed here) couldbe to base the likelihood computation only on those samples not close to zero simultaneously.

u

Rel

ativ

e er

ror

of th

e ap

prox

imat

ion

of l

og((

−1)

d ψ(d

) (t(u))

)

10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100

10−4

10−3

10−2

10−1

100

101

102

103

104

FIGURE 7. Relative error as a function of u based on m = 10000, d = 5, and θ = 2.




5.3. Gumbel’s and Joe’s polynomial

For computing the log-likelihood for the Archimedean Gumbel or Joe copula, we need an efficientway of evaluating the logarithm of the density cθ (uuu) as given in Section 3.4, Parts 4 and 5,respectively. The challenge is to evaluate the logarithm of the polynomials involved. For this thefollowing auxiliary results are essential. Their proofs are straightforward and thus omitted.

Lemma 5.1

1. Let xi ≥ 0, i ∈ {1, . . . ,n}, such that ∑ni=1 xi > 0. Furthermore, let bi = logxi, i ∈ {1, . . . ,n},

with log0 =−∞, and let bmax = max1≤i≤n bi. Then

logn

∑i=1

xi = bmax + logn

∑i=1

exp(bi−bmax). (14)

2. Let xi ∈R, i ∈ {1, . . . ,n}, such that ∑ni=1 xi > 0. Furthermore, let si = signxi, bi = log|xi|,

i ∈ {1, . . . ,n}, with log0 =−∞ and let bmax = max1≤i≤n bi. Then

logn

∑i=1

xi = bmax + log( n

∑i=1

i:si=1

exp(b(i)−bmax)−n

∑i=1

i:si=−1

exp(b(i)−bmax)

), (15)

where b(i) denotes the ith smallest value of bi, i ∈ {1, . . . ,n}.

The ideas behind Lemma 5.1 1 and 2 are implemented in the (non-exported) functions lsumand lssum in the R package copula.

Although mathematically straightforward, Lemma 5.1 has an important consequence for evalu-ating logarithms of polynomials such as PG

d,α for Gumbel’s or PJd,α for Joe’s density. Depending

on the evaluation point, it might happen that the value of the polynomial is not representablein computer arithmetic and thus one cannot first compute the value of the polynomial and takethe logarithm afterwards. Instead, Formula (14) suggests a “intelligent” (numerically stable)logarithm to compute such polynomials (or sums). By taking out the maximum of the bi, it isguaranteed that the exponentials which are summed up are all in [0,1] and thus the sum takeson values in [0,n], representable in computer arithmetic. This trick solves the numerical issuesfor computing PJ

d,α and thus for computing the log-likelihood of a Joe copula. The evaluation ofPJ

d,α is implemented as (non-exported) function polyJ in the R package copula. It is called withdefault method log.poly implementing the trick described above when evaluating the density ofa Joe copula via the slot dacopula; two other, less efficient methods are also available, one ofwhich is a straightforward polynomial evaluation (poly).

Formula (15) takes the above idea of an intelligent logarithm a step further, by dealing withpossibly negative summands. The summands in each sum are ordered in increasing order toprevent cancellation. This formula is helpful in computing PG

d,α . However, the situation turns outto be more challenging for Gumbel’s family. All in all, several different methods for the evaluationof PG

d,α were implemented. They are based on the following results about PGd,α and described

below, where here and in the following, α = 1/θ ∈ (0,1].




Lemma 5.2Let

PGd,α(x) =

d

∑k=1

aGdk(α)xk (16)

for α ∈ (0,1], where

aGdk(α) = (−1)d−k

d

∑j=k

αjs(d, j)S( j,k) (17)

=d!k!

k

∑j=1

(kj

)(α jd

)(−1)d− j, k ∈ {1, . . . ,d}. (18)

Then1. aG

dk(α) = pJ01(d;k)d!/k! for all α ∈ (0,1], where pJ

01(d;k)> 0 denotes a probability massfunction in d ∈ {k,k+1, . . .};

2. PGd,α allows for the following representations:

PGd,α(x) = (−1)d−1x

d

∑j=1

(s(d, j)

j−1

∑k=0

S( j,k+1)(−x)k)

αj (19)

= (−1)d−1αx

d−1

∑j=0

(s(d, j+1)

j−1

∑k=0

S( j,k+1)(−x)k)

αj (20)

=d

∑j=1

s j exp(log|(α j)d |+ j logx+ x− log( j!)+ logFPoi(x)(d− j)

), (21)

where, for all j ∈ {1, . . . ,d},

s j =

{(−1) j−dα je, α j /∈N or (α = 1, j = d),0, otherwise,

(22)

and where FPoi(x)(·) denotes the distribution function of a Poisson distribution with param-eter x.

ProofPart 1 of Lemma 5.2 follows from [Hofert, 2010, p. 99] (the probability mass function pJ

01(d;k)> 0corresponds to the distribution function F01(· ;k) whose Laplace-Stieltjes transform is the generatorψ01(· ;k) appearing in a nested Joe copula). In particular, this equality implies that the coefficientsaG

dk(α) of PGd,α are positive. This allows one to apply (14) with bk = logaG

dk(α)+ k logx, k ∈{1, . . . ,d} (d = n), to compute the logarithm of the polynomial aG

dk(α) at x. Concerning Part2, Equations (19) and (20) directly follow from interchanging the order of summation of (16)combined with (17). For Equation (21), note that interchanging the order of summation of (16)combined with (18) and rewriting the generalized binomial coefficient

(α jd

)as (α j)d/ j! leads to

PGd,α(x) =

d

∑j=1

(α j)d(−1)d− j x j exp(x)j!

d− j

∑k=0

xk

k!exp(−x).




Interpreting the second sum as FPoi(x)(d− j), pulling out the signs s j = sign((α j)d(−1)d− j), andbringing in an exp(log(·)) leads to the result as stated. Concerning the formula for s j, note that

s j = sign((α j)d(−1)d− j) = (−1)d− j signd−1

∏l=1

(α j− l) = (−1)d− jd−1

∏l=1

sign(α j− l)

= (−1)d− jd−1

∏l=dα je

sign(α j− l). (23)

First assume α j /∈N and dα je ≤ d−1. In this case s j = (−1)d− j(−1)d−1−dα je+1 = (−1) j−dα je.This is also true if dα je = d since then j has to be equal to d. Now consider α = 1 and j = d.In this case, it is easily seen from (23) that s j = 1 which is also true for (22). Finally, considerthe second case in (22). It implies that α j ∈N and thus the first factor in (23) being zero dueto l = dα je = α j, so s j = 0. Note the interesting fact that the formula for s j is independent ofd.

The results presented in Lemma 5.2 lead to the following different methods for computing thelogarithm of PG

d,α , see the (non-exported) function polyG of the R package copula:– pois, pois.direct: These methods are based on Representation (21), where pois applies

the intelligent logarithm (15) to evaluate the sum and pois.direct computes the sumdirectly as given in (21);

– stirling, stirling.horner: Method stirling evaluates Representation (19) directly,where the polynomial ∑

j−1k=0 S( j,k+1)(−x)k in−x is computed via Horner’s scheme. Method

stirling.horner is based on Representation (20), where Horner’s scheme is applied tocompute the polynomial in α with coefficients s(d, j+1)∑

j−1k=0 S( j, k+1)(−x)k which, as

before, are evaluated with Horner’s scheme.– sort, horner, direct, and dsSib.*: These methods all PG

d,α with the intelligent logarithm(14) based on the logarithms of the coefficients aG

dk(α) (note that the coefficients aGdk(α)

of are all positive). The logarithmic coefficients can be obtained in different ways: sortcomputes them via (15); horner via Horner’s scheme based on interpreting (17) as apolynomial in −α; direct by directly computing the sum as given in (17); and dsSib.*by various different methods described on the help page of the function dsumSibuya (forexample, dsSib.log uses (18) together with the intelligent logarithm as given in (15)).

Additionally, a method default is implemented which consists of a careful combination of theabove methods based on numerical experiments. Note that all methods involved work with logxinstead of x. For this reason, polyG requires as argument logx rather than x.

Finally, let us mention that the problem of evaluating sums of type

n

∑k=0

(nk

)(−1)k fk (24)

for sequences ( fk)k has a long history (note that (18) falls under this setup). They can be interpretedas forward differences and are known to be numerically challenging. Approximate (asymptoticfor n→ ∞) formulas may be obtained by using methods from complex analysis; see, for example,[Flajolet and Sedgewick, 1995], and have been important, e.g., for estimating the complexity ofcomputer algorithms.




However, these asymptotic formulas are not very accurate for finite n (note that in our case,n = d, the data dimension) and the only known way to accurately compute them, seems highprecision arithmetic. See sumBinomMpfr() in R package Rmpfr, and its documentation forsimple examples such as fk = f (k) =

√k.

5.4. Kendall’s tau for Ali-Mikhail-Haq copulas

In Table 1, the population version of Kendall’s tau for the Ali-Mikhail-Haq (A) family, as functionof the parameter θ , is

τA(θ) = 1−2(θ +(1−θ)2 log(1−θ))/(3θ2). (25)

When computing it for the Kendall’s tau estimator (see Section 3.1), however, the simple formula(25) is not sufficient, notably not for small θ , see the left plot in Figure 8: Replacing log(1−θ)by its numerical accurate log1p(−θ) helps down to around θ ≈ 10−7, but then that formulabreaks down as well, and indeed our tauAMH() (package copula), uses parts of the Taylor seriesτA(θ) =

29 θ(1+θ(1

4 +θ

10(1+θ(12 +θ

27))))+O(θ 6) 1, as soon as θ ≤ 10−2.

θ

τ AM

H(θ

)

10−10 10−8 10−6 10−4 10−2 100

10−11

10−9

10−7

10−5

10−3

10−1

1 − 2((1 − θ)2log(1 − θ) + θ) (3θ2)1 − 2((1 − θ)2log1p(− θ) + θ) (3θ2)tauAMH(θ)

θ

|1−

τ(θ)

τ(θ)

|

w/ log1p()

10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 10

10−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

k= 3

k= 4

k= 5

k= 6

k= 7

tauAMH()

FIGURE 8. left: τA(θ): direct form (breaking down for θ < 10−5), using log1p() (okay down to θ ≈ 10−8), correctapproximation as provided by tauAMH(); right: relative errors of log1p(), Taylor approximations, and our hybridtauAMH().

5.5. log1mexp

There are several situations, such as the one addressed in Section 5.6, where an accurate computa-tion of

f (a) = log1mexp(a) = log(1− exp(−a)), a≥ 0, (26)

1 replacing log(1 − θ) by its Taylor expansion −∑∞k=1 θ k/k in (25) results in the expansion τA(θ) =

29 θ ∑

∞k=0

6(k+1)(k+2)(k+3) θ k




is required. Note that this is numerically challenging in both situations, when a ↓ 0 (henceexp(−a) ↑ 1 and cancellation of two almost equal terms in 1− exp(−a)), and when a ↑ ∞, asexp(−a) ↓ 0 and in 1−exp(−a), almost all accuracy of exp(−a) is lost when it is less than around10−15. Now, for the first case, we can make use of the R and C library function expm1(x) whichcomputes exp(x)− 1 accurately also for very small x, and for the second case, use the R andC library function log1p(x) which computes log(1+ x) accurately also for very small x. Ourpackage copula provides the function log1mexp() which adapts to these two cases, in a sense,optimally by using a cutoff of a = log2; see [Mächler, 2012].

5.6. The density of the diagonal of Frank copulas

Computing the DMLE for Frank’s copula family is one situation where the accurate computationof (26) is crucial. To compute the density δ ′

θof the diagonal (12) for Frank copulas with generator

ψθ (t) =− log(1− (1− exp(−θ))e−t)/θ , the functions

−ψ′θ (t) =

(1− e−θ )exp(−t)θ(1− (1− e−θ )exp(−t))

,

ψ−1θ

(u) =− log(

exp(−uθ)−1e−θ −1

), and− (ψ−1

θ)′(u) =

θ

exp(θu)−1

are involved. Numerical issues in computing δ ′θ(u) arise for large θ and u close to 1. It is known

that numerically, the computation of ex−1 suffers from cancellation when 0 < x� 1. The firstsuspect is thus ψ

−1θ

which involves terms of this type. The left-hand side of Figure 9 displayslogδ ′

θ(u) for θ = 38 and d = 2 for two different versions of computing ψ

−1θ

: psiInv.0 usesonly the R functions log() and exp(), whereas psiInv.1 uses log() and expm1(). Eitherway, numerical issues appear due to the cancellation in the division of terms of type ex−1 whencomputing ψ

−1θ

. By rewriting ψ−1θ

as

ψ−1θ

(u) =− log(

1− exp(−uθ)− e−θ

1− e−θ

)

we can use R’s function log1p(.) to accurately compute log(1+ ·) and thus the generator inverseψ−1θ

via -log1p((exp(-u*theta)-exp(-theta))/expm1(-theta)) which we denote bypsiInv.2. The right-hand side of Figure 9 displays the effect of using psiInv.2 in comparisonto psiInv.0 and psiInv.1.

Although this already looks promising, it is still not possible to compute the negative log-likelihood for the DMLE of Frank’s copula family for a large range of parameters θ as one wouldlike to do for the optimization. The left-hand side of Figure 10 shows the negative log-likelihoodbased on the diagonal of a five-dimensional (so rather low-dimensional) Frank copula, wherecomputations are done in double precision and high-precision arithmetic with different significantbits (this was done with the R package Rmpfr). As it turns out, the problem is the evaluation of−ψ ′

θ(t) for small t (equivalently, t = ψ

−1θ

(u) for large θ and u close to 1 as before). The solution




0.0 0.2 0.4 0.6 0.8 1.0

−2.

5−

2.0

−1.

5−

1.0

−0.

50.

0

u

log

δ′38

(u)

for

d=

2

psiInv.0(.)psiInv.1(.)

0.0 0.2 0.4 0.6 0.8 1.0

−2

−1

01

23

u

log

δ′38

(u)

for

d=

2

psiInv.2(.)psiInv.1(.)

FIGURE 9. logδ ′θ(u) for θ = 38 and d = 2 for computing ψ

−1θ

via R’s log() and exp() functions (psiInv.0), aversion with log() and expm1() (psiInv.1), and a version using log1p() and expm1() (psiInv.2).

is to rewrite the logarithm of −ψ ′θ(t) via

log(−ψ′θ (t)) = log(1− e−θ )− t− log(θ)− log(1− (1− e−θ )exp(−t))

= log1mexp(θ)− t− log(θ)− log1mexp(− log((1− e−θ )exp(−t))

)= log1mexp(θ)− t− log(θ)− log1mexp

(t− log(1− e−θ )

)= w− log(θ)− log1mexp(−w),

where w = log1mexp(θ)− t. By computing log1mexp via log1mexp() as described in Sec-tion 5.5, one can then accurately compute the negative log-likelihood for the DMLE for Frank’scopula family; see the right-hand side of Figure 10. For more details we refer the interested readerto [Mächler, 2011] 2.

6. Conclusion

Motivated by applications in quantitative risk management, we introduced and compared differentparametric estimators for Archimedean copula families with focus on large dimensions (up tod = 100). In particular, estimators based on Kendall’s tau, Blomqvist’s beta, minimum distanceestimators, the maximum-likelihood estimator, a simulated maximum-likelihood estimator, and amaximum-likelihood estimator based on the copula diagonal were investigated both under knownand unknown margins (pseudo-observations). Several of these estimation methods were newlyintroduced and investigated in this context.

Under known margins, the best performance according to precision was shown by the maximum-likelihood estimator. To our surprise, the maximum-likelihood estimator also performed wellaccording to numerical stability (being of similar numerical stability as the pairwise Kendall’s

2 As this is a vignette of R package copula, all its figures are completely reproducible via R code in the fileFrank-Rmpfr.Rnw which is part of the package source.




20 40 60 80 100

−10

−8

−6

−4

−2

θ

−lo

gLik

(θ, .

) fo

r d

=5

and

n=

100

double precisionmpfr(*, precBits = 160)mpfr(*, precBits = 128)mpfr(*, precBits = 96)

20 40 60 80 100

−6

−5

−4

−3

−2

θ

−lo

gLik

(θ, .

) fo

r d

=5

and

n=

100

FIGURE 10. Negative log-likelihood based on the diagonal of a five-dimensional Frank copula (sample size n = 100)with various kinds of precision (left) and an accurate version in double precision based on log1mexp (right).

tau estimators) and run time (being only outperformed by the diagonal maximum likelihoodestimator). Under unknown margins, the MLE still performed best, but the differences in precisionbetween the various estimators are much less clear-cut and the rate of improvement in d is not ashigh as under known margins.

Our work specifically addressed the challenges of inference in large dimensions which isimportant for practical applications. Large dimensions up to d = 100 were tackled for the firsttime and numerical challenges when working in such large dimensions were addressed in detail.Moreover, a detailed implementation of the presented estimation methods in the R packagecopula creates transparency and allows the reader to access and verify our results.

Archimedean copulas in high dimensions are often used for modeling dependence in largeportfolios of various kinds, for example, if tail dependence is of interest or only few data isavailable. Furthermore, Archimedean copulas serve as building blocks for several extensionswhich are not limited by exchangeability and thus allow for a wider range of statistical applications.The numerical solutions studied in this paper extend to these models. Furthermore, importantstatistical quantities used for sampling or goodness-of-fit testing such as distributions of radialparts or the Kendall distribution function require accurate evaluation procedures, which are nowavailable even in high dimensions.

Appendix A: Simulation tables




TA

BL

E2.B

iases(under

known

margins)m

ultipliedby

1000.Thenum

bersin

parenthesesdenote

thefactors

ofthecorresponding

entriesw

ithrespectto

theperform

anceofthe

MLE

.

1000B

iasE

stimator

τFam

.d

τ¯τ

τ¯θ

βM

DE

CvM

χM

DE

KS

χM

DE

CvM

ΓM

DE

KS

ΓM

LE

SML

ED

ML

E

0.25A

5−

4.3(1.6)−

22.9(8.7)

−10.1

(3.9)

−12.2

(4.6)

−7.4

(2.8)

−54

.3(20.6

)−

824.7(313.7

)−

2.6

(1.0)

−7.0

(2.7)

−35

.8(13

.6)20

−5.5

(−15.2

)−

20.2(−

55.5)

−42.9

(−117

.7)

−1.9

(−5.3)

−3.8

(−10.4

)−

38.3

(−105

.0)

−831.0

(−2277

.7)

0.4

(1.0)

−0.4

(−1.0

)−

27.9(−

76.5)

1000.9

(−9.5

)−

21.9(241.5

)171

.3(−

1888.9)

−0.0

(0.5)

0.1

(−1.0)

−39

.2(432

.7)

−827.3

(9122.9)−

0.1

(1.0)

−0.6

(6.7)

−26.5

(292.4)

C5

4.4(1.2)

18.9

(5.3)

4.9

(1.4)

−2.1

(−0.6)−

185.7

(−51.5

)4.6

(1.3)

−666.0

(−184

.7)3.6

(1.0)

256.2(71

.1)

−0.6

(−0.2)

203.1

(2.1)

19.7

(13.3)

−18.6

(−12

.6)−

135.6

(−91.9

)−

258.8(−

175.4)

−12.1

(−8.2)

−663.3

(−449.5

)1.5

(1.0)

526.0

(356.4)

1.8

(1.2)

100−

1.3

(321.1)24.1

(−6023

.6)

37143.5(−

9284172.1)−

418.9

(104693.9)−

351.5(87871

.1)

−21.6

(5406.7)

−661

.9(165436

.2)−

0.0(1.0)

742.1

(−185481

.5)

−4.6

(1146.6)

F5

0.3

(0.0)

35.1

(4.6)

8.2

(1.1)

0.7

(0.1)

17.9(2.3)

6.3

(0.8)−

2363.5(−

308.0)

7.7(1.0)

5.3

(0.7)

−32

.5(−

4.2)

2010

.9(12.2

)36.5

(40.9)

−14.8

(−16.6

)3.8

(4.3)

10.1(11

.3)

9.6

(10.7)−

2357.3(−

2642.6)

0.9

(1.0)

−7.5

(−8.4

)−

127.2(−

142.6)

1002.3

(−3.4

)27.1

(−39.6)

60976.9

(−89236

.7)

−5.3

(7.8)

3.7

(−5.3

)−

38.6

(56.5)−

2356.9(3449.2

)−

0.7

(1.0)

3.3(−

4.9)−

208.3(304.8

)

G5

5.2

(2.4)

10.7

(4.9)

8.4

(3.8)

1.6

(0.7)

0.1

(0.1)

12.8(5.8)

−331

.4(−

150.7)

2.2

(1.0)

7.7

(3.5)

1.6

(0.7)

202.9

(−1.7)

10.3

(−6.2)

2.4

(−1.4)

0.3

(−0.2

)−

0.2

(0.1)

7.2

(−4.3)

−326

.2(194.4

)−

1.7(1.0)

2.0

(−1.2)

0.3

(−0.2)

1004.1

(12.0)

10.2(29

.7)−

333.3(−

969.9)

−0.2

(−0.6)

−0.1

(−0.2)

10.8

(31.4)

−324.7

(−944.9

)0.3

(1.0)

0.9

(2.5)

−0.9

(−2.7)

J5

9.0(10.5

)17.9

(20.8)

4.6(5.3)

4.9

(5.7)

3.9(4.5)

28.8

(33.5)

−594.0

(−689

.9)

0.9

(1.0)

17.8(20.6

)13.6

(15.8)

209.8

(10.2)

19.6(20

.4)

18.1

(18.9)

−0.7

(−0.8)

1.9

(2.0)

44.2

(46.1)

−591.9

(−617

.7)

1.0

(1.0)

9.9

(10.4)

4.4

(4.6)

1002.8

(8.2)

20.1(58.9

)33980

.2(99368.1)

0.4

(1.2)

0.2

(0.7)

60.2(175

.9)−

594.9(−

1739.7)0.3

(1.0)

−0.0

(−0.0)

0.7(2.2

)

0.75

C5

74.6(6.8)

164.8

(15.1)

197.1(18

.1)−

3988.0(−

365.3)−

4140.0(−

379.3)−

1632.3(−

149.5)−

6000.0

(−549.7

)10.9

(1.0)

119.7

(11.0)

719.9

(66.0)

20118

.9(−

164.8)145

.1(−

201.1)

176.7(−

245.0)−

5849.9(8109.5

)−

5521.4(7654.1

)−

1714.1(2376.2

)−

5999.8(8317.3

)−

0.7(1.0)

99.1

(−137.4

)281

.3(−

389.9)

10024

.1(20

.4)

106.0(89

.8)

31810.2(26967

.5)−

5976.2(−

5066.4)−

5946.2(−

5041.0)−

1650.7

(−1399.4

)−

5996.4(−

5083.5)

1.2(1.0)

61.2

(51.8)

161.5(136

.9)

F5

82.2

(5.3)

138.2(8.9)

532.9(34.3

)25.4

(1.6)

−3.9

(−0.3)

161.0

(10.4)−

14089.0

(−906

.6)

15.5

(1.0)

195.6(12.6

)643.6

(41.4)

20−

0.7(−

0.0)

201.5(11

.3)

260.1(14

.6)

9.6

(0.5)

−7.6

(−0.4)

95.8

(5.4)−

14043.0

(−786

.5)

17.9

(1.0)

77.9(4.4

)−

58.3(−

3.3)

10031

.9(−

6.0)

173.8(−

32.8)

49210.4

(−9293.0

)−

2.3(0.4)

8.2

(−1.6

)37

.8(−

7.1)−

14057.7

(2654.7)−

5.3

(1.0)

24.1(−

4.6)−

194.7(36.8

)

G5

57.0

(40.8)

42.2

(30.3)

140.8

(100.9)

3.7

(2.7)

−0.3

(−0.2

)27.3

(19.5)

−2998.7

(−2148.7)

1.4

(1.0)−

15.3(−

11.0)

199.0(142.6

)20

16.1

(−9.9)

68.5

(−42.2

)52.3

(−32

.2)1.5

(−0.9)

2.1

(−1.3)

40.3(−

24.8)−

2996.6

(1843.4)−

1.6

(1.0)−

11.4

(7.0)

53.6

(−33

.0)

10019

.6(−

14.8)

57.5

(−43.5)−

2999.9(2268.3

)2.0

(−1.5

)−

0.2

(0.2)

30.2

(−22.8

)−

2997.8(2266.6

)−

1.3(1.0)

−9.9

(7.5)

3.4

(−2.6)

J5

24.0(1.3)

123.7(6.7)

236.8(12.9

)15.2

(0.8)

−0.6

(−0.0)

105.2

(5.7)−

5781.1(−

314.9)

18.4

(1.0)−

27.2

(−1.5

)219.0

(11.9)

2086

.0(15.9

)133.4

(24.7)

82.0(15

.2)

4.4

(0.8)

6.8

(1.3)

98.6

(18.3)−

5776.6(−

1070.5)

5.4

(1.0)

−0.4

(−0.1)

−0.1

(−0.0)

100109

.8(33

.0)

121.0(36.3

)28759

.0(8635.4)

0.2

(0.1)

0.2

(0.1)

63.5(19

.1)−

5782.3(−

1736.2)3.3

(1.0)

8.8(2.7)

−18

.1(−

5.4)




TAB

LE

3.R

MSE

(und

erkn

own

mar

gins

)mul

tiplie

dby

1000

.The

num

bers

inpa

rent

hese

sde

note

the

fact

ors

ofth

eco

rres

pond

ing

entr

ies

with

resp

ectt

oth

epe

rfor

man

ceof

the

MLE

.

1000

RM

SEE

stim

ator

τFa

mily

dτ

¯ ττ

¯ θβ

MD

EC

vMχ

MD

EK

Sχ

MD

EC

vMΓ

MD

EK

SΓ

ML

ESM

LE

DM

LE

0.25

A5

77.2

(1.7)

81.5

(1.8)

125.

3(2.7)

80.8

(1.8)

80.4

(1.8)

213.

2(4.6)

831.

5(1

8.1)

45.9

(1.0)

48.2

(1.0)

115.

7(2.5)

2051.5

(3.1)

54.4

(3.3)

160.

1(9.7)

35.3

(2.1)

37.5

(2.3)

167.

7(1

0.2)

834.

3(5

0.7)

16.5

(1.0)

17.5

(1.1)

84.3

(5.1)

100

45.9

(10.

0)49.4

(10.

8)17

1.3

(37.

4)15

.5(3.4)

16.1

(3.5)

158.

7(3

4.6)

832.

6(1

81.6)

4.6

(1.0)

5.1

(1.1)

80.6

(17.

6)

C5

138.

6(1.8)

139.

7(1.8)

168.

3(2.2)

120.

6(1.6)

367.

5(4.8)

281.

6(3.7)

666.

3(8.7)

76.8

(1.0)

2127.0

(27.

7)21

7.4

(2.8)

2010

0.1

(2.9)

100.

6(3.0)

172.

3(5.1)

306.

1(9.0)

416.

3(1

2.3)

262.

8(7.7)

666.

0(1

9.6)

33.9

(1.0)

2881.6

(84.

9)16

1.2

(4.7)

100

89.0

(6.1)

97.7

(6.6)

3714

3.5

(252

5.6)

528.

5(3

5.9)

484.

3(3

2.9)

224.

8(1

5.3)

663.

7(4

5.1)

14.7

(1.0)

3563

.2(2

42.3)

147.

1(1

0.0)

F5

349.

5(1.2)

350.

8(1.3)

497.

6(1.8)

398.

4(1.4)

406.

4(1.5)

1028

.9(3.7)

2368.7

(8.5)

279.

6(1.0)

292.

2(1.0)

623.

4(2.2)

2025

1.5

(2.0)

251.

6(2.0)

538.

4(4.3)

185.

1(1.5)

191.

4(1.5)

961.

1(7.7)

2364.4

(18.

9)12

5.3

(1.0)

129.

4(1.0)

423.

4(3.4)

100

217.

4(5.0)

218.

3(5.0)

6097

6.9

(139

9.6)

86.7

(2.0)

82.0

(1.9)

903.

3(2

0.7)

2364.0

(54.

3)43.6

(1.0)

50.2

(1.2)

398.

9(9.2)

G5

71.0

(1.5)

73.2

(1.5)

96.7

(2.0)

59.0

(1.2)

57.6

(1.2)

165.

7(3.5)

332.

7(7.0)

47.7

(1.0)

50.4

(1.1)

109.

2(2.3)

2057.8

(2.5)

62.0

(2.7)

110.

1(4.7)

26.8

(1.2)

28.0

(1.2)

150.

6(6.5)

335.

3(1

4.5)

23.2

(1.0)

23.8

(1.0)

59.1

(2.6)

100

57.9

(5.6)

57.7

(5.6)

333.

3(3

2.3)

12.6

(1.2)

12.5

(1.2)

145.

8(1

4.1)

338.

2(3

2.8)

10.3

(1.0)

12.3

(1.2)

37.6

(3.6)

J5

139.

5(1.7)

146.

9(1.8)

165.

5(2.0)

98.9

(1.2)

104.

7(1.3)

286.

8(3.5)

596.

5(7.3)

81.6

(1.0)

90.5

(1.1)

185.

2(2.3)

2013

2.2

(3.6)

126.

4(3.4)

214.

4(5.8)

46.1

(1.2)

46.4

(1.2)

306.

7(8.3)

597.

0(1

6.1)

37.2

(1.0)

47.9

(1.3)

92.3

(2.5)

100

124.

2(8.2)

126.

7(8.3)

3398

0.2

(223

7.0)

21.4

(1.4)

21.6

(1.4)

352.

1(2

3.2)

595.

4(3

9.2)

15.2

(1.0)

22.5

(1.5)

53.7

(3.5)

0.75

C5

871.

6(3.1)

869.

8(3.1)

1456

.9(5.1)

4897

.1(1

7.2)

4983.8

(17.

5)32

34.4

(11.

3)60

00.0

(21.

0)28

5.2

(1.0)

1430.2

(5.0)

3545.5

(12.

4)20

770.

1(6.0)

773.

6(6.0)

1040.2

(8.1)

5924.6

(46.

0)57

55.2

(44.

7)32

96.7

(25.

6)59

99.8

(46.

6)12

8.7

(1.0)

1284.5

(10.

0)17

06.7

(13.

3)10

069

8.6

(12.

4)72

2.2

(12.

8)31

810.

2(5

63.2)

5988.0

(106

.0)

5972.9

(105.7)

3221.1

(57.

0)59

97.2

(106.2)

56.5

(1.0)

1019.0

(18.

0)11

72.5

(20.

8)

F5

1014

.1(1.5)

1005.7

(1.5)

3232.6

(4.8)

766.

6(1.1)

807.

3(1.2)

2124

.4(3.2)

1409

5.8

(21.

1)66

8.6

(1.0)

793.

1(1.2)

4105.2

(6.1)

2070

2.8

(2.4)

760.

2(2.6)

2169.0

(7.3)

354.

9(1.2)

361.

7(1.2)

2081.9

(7.0)

1406

7.2

(47.

3)29

7.2

(1.0)

347.

8(1.2)

2023.9

(6.8)

100

639.

0(4.9)

692.

9(5.3)

4921

0.4

(380.0)

158.

2(1.2)

160.

9(1.2)

1943.9

(15.

0)14

071.

0(1

08.6)

129.

5(1.0)

147.

7(1.1)

1273.0

(9.8)

G5

385.

1(2.4)

383.

6(2.4)

763.

5(4.8)

173.

0(1.1)

177.

2(1.1)

462.

3(2.9)

2998.8

(18.

8)15

9.7

(1.0)

185.

5(1.2)

1238.1

(7.8)

2033

9.4

(4.7)

337.

0(4.7)

509.

8(7.1)

80.8

(1.1)

83.3

(1.2)

471.

9(6.5)

2997.0

(41.

5)72

.2(1.0)

105.

1(1.5)

568.

8(7.9)

100

327.

3(1

0.2)

339.

3(1

0.6)

2999.9

(93.

9)37

.4(1.2)

38.5

(1.2)

452.

8(1

4.2)

2998

.2(9

3.9)

31.9

(1.0)

77.8

(2.4)

353.

3(1

1.1)

J5

853.

2(2.9)

851.

6(2.8)

1366.3

(4.6)

341.

0(1.1)

341.

5(1.1)

947.

5(3.2)

5781.2

(19.

3)29

9.2

(1.0)

369.

0(1.2)

1821.4

(6.1)

2078

5.7

(6.0)

823.

0(6.3)

874.

6(6.7)

156.

9(1.2)

158.

3(1.2)

910.

4(7.0)

5777.9

(44.

2)13

0.6

(1.0)

185.

3(1.4)

757.

8(5.8)

100

751.

6(1

2.7)

784.

0(1

3.2)

2878

0.2

(485.8)

73.5

(1.2)

72.6

(1.2)

922.

3(1

5.6)

5782.3

(97.

6)59.2

(1.0)

127.

3(2.1)

445.

9(7.5)




TA

BL

E4.M

eanuser

runtim

es(under

known

margins)in

milliseconds.The

numbers

inparentheses

denotethe

factorsofthe

correspondingentries

with

respecttothe

performance

oftheM

LE.

Usertim

ein

ms

Estim

ator

τFam

ilyd

τ¯τ

τ¯θ

βM

DE

CvM

χM

DE

KS

χM

DE

CvM

ΓM

DE

KS

ΓM

LE

SML

ED

ML

E

0.25A

56.4

(0.3)

10.2(0.5)

3.1

(0.2)

64.7

(3.3)

90.1

(4.6)

70.1

(3.6)

96.5

(4.9)

19.5

(1.0)

3950.4(202.1

)3.4

(0.2)

2058.6

(2.3)

142.5(5.6

)3.8

(0.1)

277.5

(11.0)

389.7

(15.4)

299.1

(11.8)

416.0

(16.5)

25.2(1.0

)3598.9

(142.6)

3.8

(0.2)

1001340.6

(19.1)

3498.9(49.7

)6.8

(0.1)

2511.6(35

.7)

3534.4(50

.2)

3272.5(46

.5)

4743.4(67

.4)

70.4(1.0

)4202

.8(59.7

)4.7

(0.1)

C5

5.7(0.3)

5.7

(0.3)

2.3(0.1)

225.4(9.9)

241.3(10

.6)212.0

(9.3)

157.1(6.9)

22.8

(1.0)

10362.1

(454.7)

2.8(0.1)

2058.2

(2.0)

58.2

(2.0)

2.8

(0.1)

958.9(33

.1)

887.9(30.6

)963.9

(33.3)

666.7(23.0

)29

.0(1.0)

10179.1

(351.2)

2.4

(0.1)

1001342

.1(18

.5)

1344.8(18

.5)6.6

(0.1)

8379.0(115.4

)8890.5

(122.4)

10115.4

(139.3)

6551.3(90.2

)72

.6(1.0)

10425.8

(143.6)

3.3

(0.0)

F5

7.3

(0.3)

13.8(0.5)

5.0

(0.2)

172.2(6.6)

201.3(7.8)

180.2(6.9)

196.9(7.6)

26.0

(1.0)

11106.5

(427.7)

11.7

(0.5)

2059.2

(1.4)

218.8(5.1)

4.9

(0.1)

777.9(18.2

)919.1

(21.5)

729.7(17.0

)883.8

(20.6)

42.8

(1.0)

11860.2

(277.0)

12.1

(0.3)

1001346.3

(10.9)

5289.8(42.8

)10.0

(0.1)

7247.7(58.7

)8489

.3(68.7

)7266.4

(58.8)

8234.5

(66.6)

123.6

(1.0)

11279.1

(91.3)

12.0

(0.1)

G5

6.0

(0.1)

5.9

(0.1)

2.4

(0.0)

45.0(0.6)

79.1

(1.0)

52.3

(0.7)

85.8

(1.1)

78.4

(1.0)

10081.0

(128.5)

1.4

(0.0)

2058.5

(0.5)

58.6(0.5

)2.8

(0.0)

226.7

(1.9)

314.2

(2.6)

225.3

(1.9)

423.3

(3.5)

119.8(1.0)

9596.7(80.1

)1.8

(0.0)

1001336.6

(3.6)

1341.9(3.6

)4.4

(0.0)

2657.2

(7.1)

3701.2(9.9)

2617.7(7.0

)4376.9

(11.7)

373.7(1.0

)9739.1

(26.1)

2.5

(0.0)

J5

7.2(0.2)

16.5(0.4

)4.6

(0.1)

48.5

(1.2)

85.0

(2.1)

56.8

(1.4)

92.1

(2.2)

41.3(1.0

)11156

.2(270.1

)6.3

(0.2)

2059.6

(1.0)

331.9

(5.5)

4.8(0.1)

237.7(4.0)

311.1(5.2)

245.2(4.1)

402.1(6.7)

59.9

(1.0)

11234.0

(187.4)

8.0(0.1)

1001377

.6(9.1)

8047.2(53

.1)10

.0(0.1)

2198.1(14

.5)3486.4

(23.0)

2802.5(18

.5)4158.8

(27.5)

151.5

(1.0)

9975.0(65

.9)

9.0

(0.1)

0.75

C5

5.6(0.3)

5.8

(0.3)

2.8

(0.2)

175.7(10.1

)177.0

(10.2)

216.9(12.5

)158.3

(9.1)

17.3

(1.0)

10301.2

(593.8)

2.6

(0.1)

2056.0

(2.2)

56.5(2.2)

3.0

(0.1)

636.4(25.3

)674

.6(26.8

)940.4

(37.3)

630.2

(25.0)

25.2

(1.0)

10317.8

(409.4)2.4

(0.1)

1001275.8

(18.0)

1280.0(18.1

)7.0

(0.1)

7541.1

(106.4)

6855.5

(96.8)

9827.8

(138.7)

7086.2(100.0

)70

.8(1.0)

9793.6(138.2

)3.4

(0.0)

F5

6.7

(0.3)

16.2(0.7)

4.9

(0.2)

201.8

(8.5)

203.6

(8.5)

210.8

(8.8)

217.1

(9.1)

23.9

(1.0)

11028.6

(462.4)

10.9

(0.5)

2056.9

(1.6)

247.0(6.8

)5.2

(0.1)

847.6

(23.2)

905.7

(24.8)

841.4

(23.1)

796.7

(21.8)

36.5(1.0

)11474

.2(314.6

)11

.4(0.3)

1001306.4

(15.2)

6880.2(79.9

)9.4

(0.1)

8067.5(93

.7)

8728.4(101

.4)

8501.9(98

.8)

8625.5(100

.2)

86.1(1.0

)10596

.2(123.1

)12.8

(0.1)

G5

5.7

(0.1)

5.8

(0.1)

2.5(0.0)

60.0

(1.1)

74.9

(1.4)

52.0

(1.0)

90.2(1.7)

53.1

(1.0)

9570.5(180

.3)

1.4(0.0)

2056.0

(0.6)

56.4

(0.6)

2.9

(0.0)

248.6(2.8)

313.8(3.5)

269.7(3.0)

401.0(4.5)

88.5

(1.0)

9368.8(105

.8)1.7

(0.0)

1001279.3

(4.4)

1275.5(4.4)

4.7

(0.0)

2924.9(10.0

)3379

.0(11.5

)2705.2

(9.2)

4126.4

(14.1)

292.6

(1.0)

9533.3(32

.6)2.4

(0.0)

J5

7.3(0.2)

18.4(0.6)

4.8

(0.2)

64.6

(2.0)

97.7

(3.1)

66.6

(2.1)

103.1

(3.3)

31.7

(1.0)

10136.8

(320.2)

5.5

(0.2)

2057.4

(1.3)

357.9(8.3

)6.2

(0.1)

308.9

(7.2)

371.5

(8.6)

289.5

(6.7)

431.1

(10.0)

43.0

(1.0)

10660.1

(247.7)

7.7

(0.2)

1001279.2

(9.9)

8342.0(64.9

)9.9

(0.1)

2921.9

(22.7)

4094.5(31

.8)

3137.8(24

.4)

4657.5(36

.2)

128.6(1.0

)9835

.1(76.5

)8.9

(0.1)




TAB

LE

5.B

iase

s(u

nder

unkn

own

mar

gins

)mul

tiplie

dby

1000

.The

num

bers

inpa

rent

hese

sde

note

the

fact

ors

ofth

eco

rres

pond

ing

entr

ies

with

resp

ectt

oth

epe

rfor

man

ceof

the

MLE

.

1000

Bia

sE

stim

ator

τFa

m.

dτ

¯ ττ

¯ θβ

MD

EC

vMχ

MD

EK

Sχ

MD

EC

vMΓ

MD

EK

SΓ

ML

ESM

LE

DM

LE

0.25

A5

−4.

3(−

1.6)

−22.9

(−8.

4)−

13.8

(−5.

1)34.9

(12.

8)41.8

(15.

3)10

.4(3.8)

−82

8.8

(−30

3.2)

2.7

(1.0)

−1.

3(−

0.5)

−33

.7(−

12.3)

20−

1.9

(−0.

6)−

20.1

(−6.

1)−

64.5

(−19

.6)

48.5

(14.

7)48.1

(14.

6)9.

2(2.8)

−82

6.7

(−25

1.1)

3.3

(1.0)

4.5

(1.4)

−32.4

(−9.

8)10

0−

0.5

(0.1)−

22.3

(3.6)

125.

2(−

20.0)

50.6

(−8.

1)53

.1(−

8.5)

6.1

(−1.

0)−

825.

7(1

31.6)−

6.3

(1.0)

−5.

8(0.9)

−37.9

(6.0)

C5

4.4

(0.1)

18.9

(0.4)

1.5

(0.0)

84.2

(1.7)

82.8

(1.7)

61.8

(1.3)

−65

7.5

(−13

.5)

48.8

(1.0)

58.0

(1.2)

−33.4

(−0.

7)20

5.3

(0.1)

24.1

(0.5)

−24.7

(−0.

6)87

.5(2.0)

92.5

(2.1)

49.9

(1.1)

−63

3.3

(−14

.2)

44.6

(1.0)

42.3

(0.9)

−44

.6(−

1.0)

100

9.9

(0.2)

20.3

(0.4)

3714

3.5

(779.0)

85.9

(1.8)

84.3

(1.8)

20.6

(0.4)

−63

8.4

(−13.4)

47.7

(1.0)

43.1

(0.9)

−55

.0(−

1.2)

F5

0.3

(0.0)

35.1

(1.5)

−17.1

(−0.

7)23

4.6

(10.

1)25

8.9

(11.

1)17

3.6

(7.5)−

2358.6

(−10

1.5)

23.2

(1.0)

31.4

(1.4)

−98

.4(−

4.2)

201.

3(0.0)

29.9

(1.1)

−77.1

(−2.

9)28

5.2

(10.

7)29

0.9

(10.

9)23

1.8

(8.7)−

2345.6

(−87.9)

26.7

(1.0)

12.9

(0.5)−

174.

2(−

6.5)

100

17.0

(−2.

3)39.6

(−5.

4)60

976.

9(−

8308

.9)

326.

8(−

44.5)

310.

1(−

42.3)

252.

8(−

34.4)−

2367.5

(322

.6)−

7.3

(1.0)

2.6

(−0.

4)−

163.

0(2

2.2)

G5

5.2

(0.2)

10.7

(0.4)

3.1

(0.1)

39.1

(1.3)

37.0

(1.2)

33.4

(1.1)

−33

2.0

(−10

.9)

30.4

(1.0)

38.1

(1.3)

14.1

(0.5)

201.

7(0.1)

10.1

(0.3)

−0.

9(−

0.0)

40.2

(1.2)

42.9

(1.3)

43.5

(1.3)

−32

9.1

(−9.

8)33

.5(1.0)

37.8

(1.1)

24.5

(0.7)

100

1.8

(0.0)

9.9

(0.2)

−33

3.3

(−7.

9)43

.1(1.0)

45.7

(1.1)

54.4

(1.3)

−31

8.0

(−7.

6)42.0

(1.0)

43.3

(1.0)

60.2

(1.4)

J5

7.2

(0.2)

20.8

(0.5)

7.7

(0.2)

67.9

(1.6)

62.6

(1.5)

69.7

(1.6)

−59

2.7

(−13.8)

43.1

(1.0)

50.1

(1.2)

32.8

(0.8)

205.

7(0.2)

18.9

(0.5)

4.0

(0.1)

63.3

(1.8)

68.6

(2.0)

93.4

(2.7)

−59

4.1

(−17.1)

34.7

(1.0)

37.0

(1.1)

39.7

(1.1)

100

−0.

8(−

0.0)

21.1

(0.8)

3398

0.2

(127

8.4)

71.0

(2.7)

68.9

(2.6)

104.

8(3.9)

−59

3.1

(−22

.3)

26.6

(1.0)

24.9

(0.9)

67.1

(2.5)

0.75

C5

74.6

(−0.

2)16

4.8

(−0.

5)65.6

(−0.

2)−

9.3

(0.0)

−14

.0(0.0)−

170.

4(0.5)−

5959.7

(17.

4)−

343.

0(1.0)−

310.

4(0.9)−

358.

3(1.0)

2074

.4(−

0.1)

121.

8(−

0.2)

−12

1.2

(0.2)

−88

.6(0.2)−

104.

0(0.2)−

342.

3(0.6)−

5942.6

(11.

1)−

536.

3(1.0)−

535.

6(1.0)−

681.

6(1.3)

100

37.8

(−0.

1)12

7.7

(−0.

2)31

810.

2(−

54.2)

−44

.6(0.1)

−98

.5(0.2)−

591.

6(1.0)−

5931

.5(1

0.1)−

587.

4(1.0)−

614.

7(1.0)−

708.

5(1.2)

F5

82.2

(−0.

2)13

8.2

(−0.

4)81.6

(−0.

2)34

.4(−

0.1)

39.2

(−0.

1)−

115.

8(0.3)−

1411

5.6

(39.

2)−

360.

5(1.0)−

292.

8(0.8)−

883.

0(2.4)

2040

.3(−

0.1)

171.

8(−

0.3)

−17

5.5

(0.3)

−42

.7(0.1)

−18.6

(0.0)−

373.

4(0.7)−

1405

2.1

(25.

4)−

554.

3(1.0)−

469.

4(0.8)−

695.

1(1.3)

100

−0.

8(0.0)

147.

8(−

0.3)

4921

0.4

(−87.5)−

158.

8(0.3)−

129.

6(0.2)−

546.

4(1.0)−

1408

5.0

(25.

0)−

562.

6(1.0)−

533.

3(0.9)−

537.

9(1.0)

G5

16.0

(−0.

4)52.1

(−1.

2)51.5

(−1.

2)−

7.8

(0.2)

4.7

(−0.

1)−

21.2

(0.5)−

2997.9

(70.

0)−

42.8

(1.0)−

74.0

(1.7)

−85.2

(2.0)

2030

.5(−

0.3)

67.4

(−0.

6)−

6.8

(0.1)

−36.3

(0.3)

−51

.7(0.5)−

107.

9(1.0)−

2994.4

(28.

4)−

105.

4(1.0)−

103.

6(1.0)

−1.

9(0.0)

100

32.5

(−0.

2)48.0

(−0.

4)−

2999.9

(22.

2)−

66.9

(0.5)

−69

.2(0.5)−

174.

5(1.3)−

2994.4

(22.

1)−

135.

4(1.0)−

127.

6(0.9)

42.1

(−0.

3)

J5

83.3

(−0.

3)10

6.6

(−0.

4)16

6.7

(−0.

6)−

40.2

(0.1)

−58

.0(0.2)

−89.0

(0.3)−

5779

.8(2

0.0)−

289.

6(1.0)−

290.

1(1.0)

7.6

(−0.

0)20

40.4

(−0.

1)12

2.1

(−0.

3)15.9

(−0.

0)−

117.

4(0.3)−

117.

1(0.3)−

234.

0(0.5)−

5781.6

(13.

5)−

428.

3(1.0)−

421.

0(1.0)

19.7

(−0.

0)10

098

.2(−

0.2)

171.

2(−

0.3)

2851

4.8

(−53.6)

−74

.0(0.1)−

119.

7(0.2)−

357.

3(0.7)−

5782.2

(10.

9)−

532.

0(1.0)−

487.

3(0.9)

91.5

(−0.

2)




TA

BL

E6.R

MSE

(underunknow

nm

argins)multiplied

by1000.The

numbers

inparentheses

denotethe

factorsofthe

correspondingentries

with

respecttothe

performance

oftheM

LE.

1000R

MSE

Estim

ator

τFam

ilyd

τ¯τ

τ¯θ

βM

DE

CvM

χM

DE

KS

χM

DE

CvM

ΓM

DE

KS

ΓM

LE

SML

ED

ML

E

0.25A

577

.2(1.5)

81.5(1.5

)127

.9(2.4)

70.7(1.3)

74.6

(1.4)

101.4

(1.9)

833.8(15

.7)

53.0(1.0)

53.0

(1.0)

93.7(1.8)

2052.3

(1.9)

54.3

(2.0)

185.6

(6.7)

60.8

(2.2)

59.7

(2.2)

89.5

(3.2)

832.2(30

.2)27.6

(1.0)

29.3

(1.1)

83.5(3.0)

10044.5

(2.0)

49.2

(2.2)

125.2

(5.6)

58.5

(2.6)

60.1

(2.7)

93.1

(4.2)

831.3(37

.4)

22.2

(1.0)

24.7(1.1)

75.8(3.4

)

C5

138.6(1.0

)139

.7(1.0)

180.0(1.3

)163

.9(1.2)

160.9(1.2)

207.0(1.5)

665.6(4.9)

136.9

(1.0)

149.8(1.1)

160.0(1.2

)20

98.1(0.9

)105

.9(0.9)

172.5(1.5)

133.8(1.2)

135.4(1.2)

184.4(1.6

)659

.2(5.7)

115.0(1.0)

112.8(1.0)

137.4

(1.2)

10098.2

(0.9)

96.5

(0.8)

37143.5

(324.7)

126.7(1.1)

124.7(1.1)

178.8(1.6

)655

.0(5.7

)114.4

(1.0)

110.7

(1.0)

128.1

(1.1)

F5

349.5

(1.1)

350.8(1.1)

494.9(1.5)

423.6(1.3)

444.1(1.4)

603.3

(1.8)

2366.1(7.2

)328.4

(1.0)

332.8

(1.0)

487.3

(1.5)

20243.9

(1.1)

253.2(1.1)

486.0(2.1)

366.2(1.6)

374.3(1.6)

589.9

(2.6)

2358.4(10

.2)

230.4(1.0

)235

.8(1.0)

411.1(1.8)

100215.0

(1.0)

222.4(1.0

)60976.9

(276.4)

385.4

(1.7)

372.8

(1.7)

705.9

(3.2)

2369.2(10

.7)220.6

(1.0)

219.5(1.0)

359.0(1.6)

G5

71.0

(0.9)

73.2

(0.9)

96.6

(1.2)

89.2

(1.1)

85.8

(1.0)

113.3(1.4)

332.9(4.0)

82.9

(1.0)

83.6(1.0)

88.2(1.1

)20

61.7(0.9

)60.5

(0.9)

98.9

(1.4)

76.9

(1.1)

78.8

(1.1)

117.5(1.7)

336.3(4.9)

68.8

(1.0)

73.1(1.1)

84.1(1.2

)100

55.3(0.8

)60.2

(0.8)

333.3(4.6)

75.8

(1.0)

77.1(1.1)

132.4(1.8

)350

.3(4.8)

72.8

(1.0)

74.4(1.0)

108.6

(1.5)

J5

137.4(0.9)

143.5(1.0)

178.1(1.2)

165.6(1.1)

166.8(1.1)

202.9

(1.3)

595.1

(3.9)

150.8(1.0)

159.4

(1.1)

162.4

(1.1)

20126.0

(0.9)

129.2(0.9)

179.2(1.3)

149.8(1.1)

152.3(1.1)

228.2

(1.7)

595.4

(4.4)

136.7(1.0

)138

.2(1.0

)157

.0(1.1)

100119.6

(1.0)

126.3(1.1)

33980.2(286.8

)152.5

(1.3)

149.0

(1.3)

270.0

(2.3)

596.7

(5.0)

118.5(1.0

)123

.4(1.0)

174.6(1.5)

0.75

C5

871.6

(1.1)

869.8(1.1

)1468

.2(1.8)

739.8(0.9)

756.9

(0.9)

868.2

(1.1)

5968.9(7.4)

804.3(1.0)

764.2(1.0)

1101.4(1.4)

20754.7

(0.9)

788.9

(1.0)

1043.1(1.3

)658

.6(0.8

)651

.6(0.8)

897.7(1.1)

5953.8(7.2)

826.4

(1.0)

842.4(1.0)

1043.6(1.3

)100

744.6(0.9

)723

.6(0.9)

31810.2(37

.8)

660.3(0.8)

672.4(0.8)

1038.5(1.2)

5943.0(7.1)

842.5

(1.0)

887.1(1.1)

1024.2

(1.2)

F5

1014.1

(1.0)

1005.7(1.0)

3215.4(3.2)

908.2(0.9)

940.7(0.9)

1363.9(1.4

)14118

.1(14.2

)991.5

(1.0)

1017.7

(1.0)

2024.5(2.0)

20746.2

(0.9)

756.3(0.9)

2035.2(2.4)

622.9(0.7)

605.7

(0.7)

1420.8(1.7)

14066.0

(16.4)

855.8(1.0

)838

.6(1.0)

1691.0(2.0)

100626.4

(0.7)

654.6(0.8

)49210

.4(58.1

)662

.2(0.8

)606

.7(0.7

)1580.1

(1.9)

14091.2

(16.6)

846.5(1.0)

868.7(1.0)

1415.8(1.7)

G5

379.0

(1.1)

380.4

(1.1)

757.3

(2.2)

334.1

(1.0)

330.8

(1.0)

416.8(1.2)

2998.3(8.9)

337.4

(1.0)

349.6(1.0)

414.2(1.2

)20

332.6(1.1

)337

.7(1.1)

528.0(1.7

)305

.6(1.0)

316.7(1.0)

407.8(1.3)

2995.1(9.6)

312.5

(1.0)

300.5(1.0)

345.4(1.1

)100

319.7(1.0

)318

.4(1.0)

2999.9(9.7)

307.8(1.0)

309.1(1.0)

435.7(1.4

)2995

.4(9.7

)310.0

(1.0)

307.1(1.0)

337.0

(1.1)

J5

860.7(1.1)

853.7(1.1)

1611.9(2.1)

770.0(1.0)

760.6(1.0)

906.1

(1.2)

5780.2(7.5

)771.5

(1.0)

741.5

(1.0)

1226.6(1.6)

20775.1

(1.0)

770.2(1.0)

1149.1(1.5)

687.8(0.9)

707.2

(0.9)

867.3

(1.1)

5781.6(7.5)

770.5(1.0

)769

.9(1.0)

1000.3(1.3)

100768.6

(1.0)

767.2(1.0

)28683

.7(35.5

)711

.4(0.9

)680

.8(0.8

)950

.6(1.2)

5782.2(7.2)

807.0(1.0)

778.3(1.0)

950.6(1.2)




TAB

LE

7.M

ean

user

run

times

(und

erun

know

nm

argi

ns)i

nm

illis

econ

ds.T

henu

mbe

rsin

pare

nthe

ses

deno

teth

efa

ctor

sof

the

corr

espo

ndin

gen

trie

sw

ithre

spec

tto

the

perf

orm

ance

ofth

eM

LE.

Use

rtim

ein

ms

Est

imat

or

τFa

mily

dτ

¯ ττ

¯ θβ

MD

EC

vMχ

MD

EK

Sχ

MD

EC

vMΓ

MD

EK

SΓ

ML

ESM

LE

DM

LE

0.25

A5

6.5

(0.3)

10.1

(0.5)

3.1

(0.1)

63.0

(3.0)

87.7

(4.1)

68.3

(3.2)

90.9

(4.3)

21.3

(1.0)

3966.3

(186.0)

3.5

(0.2)

2059.0

(2.4)

139.

6(5.6)

4.1

(0.2)

253.

8(1

0.2)

404.

9(1

6.3)

275.

9(1

1.1)

446.

4(1

8.0)

24.8

(1.0)

3685.1

(148.6)

3.9

(0.2)

100

1339.5

(23.

0)35

02.7

(60.

2)7.

1(0.1)

2666.4

(45.

8)38

16.6

(65.

6)33

36.2

(57.

4)42

71.6

(73.

4)58.2

(1.0)

3964

.5(6

8.2)

4.4

(0.1)

C5

6.1

(0.3)

6.1

(0.3)

2.5

(0.1)

59.7

(3.0)

84.6

(4.2)

60.1

(3.0)

90.6

(4.5)

20.1

(1.0)

1027

0.0

(510

.2)

2.9

(0.1)

2058.3

(2.2)

58.3

(2.2)

3.3

(0.1)

251.

5(9.6)

333.

9(1

2.7)

273.

8(1

0.4)

410.

7(1

5.7)

26.2

(1.0)

9752.5

(371

.7)

2.7

(0.1)

100

1336.1

(18.

2)13

37.0

(18.

2)6.

7(0.1)

2343.2

(31.

9)35

81.7

(48.

8)27

97.5

(38.

1)45

90.4

(62.

5)73

.4(1.0)

9413.1

(128.2)

3.3

(0.0)

F5

6.9

(0.2)

13.6

(0.5)

4.4

(0.2)

79.6

(2.9)

115.

8(4.2)

91.8

(3.3)

126.

6(4.5)

27.9

(1.0)

1089

5.9

(391.0)

13.3

(0.5)

2059.0

(1.4)

224.

7(5.2)

5.3

(0.1)

317.

8(7.4)

438.

5(1

0.2)

343.

8(8.0)

496.

9(1

1.5)

43.1

(1.0)

1090

8.1

(253.0)

11.7

(0.3)

100

1345.8

(11.

4)55

65.1

(47.

1)9.

0(0.1)

3325

.2(2

8.1)

4142.6

(35.

0)34

12.2

(28.

9)45

76.6

(38.

7)11

8.2

(1.0)

1120

7.6

(94.

8)12.3

(0.1)

G5

5.7

(0.1)

6.1

(0.1)

2.4

(0.0)

41.1

(0.5)

80.8

(1.0)

49.3

(0.6)

89.9

(1.1)

78.6

(1.0)

9999.5

(127

.2)

1.5

(0.0)

2058.0

(0.4)

58.4

(0.4)

2.7

(0.0)

205.

1(1.5)

355.

8(2.6)

240.

0(1.8)

387.

3(2.9)

135.

3(1.0)

9642.7

(71.

3)1.

6(0.0)

100

1367.5

(3.6)

1377.2

(3.6)

4.3

(0.0)

2222.0

(5.8)

3302

.3(8.7)

2665.2

(7.0)

4455

.5(1

1.7)

380.

7(1.0)

9115.7

(23.

9)2.

3(0.0)

J5

6.7

(0.2)

16.4

(0.4)

4.3

(0.1)

52.8

(1.2)

81.6

(1.9)

54.3

(1.3)

94.2

(2.2)

42.5

(1.0)

1062

0.3

(249.9)

5.5

(0.1)

2059.2

(1.1)

312.

0(5.6)

4.5

(0.1)

227.

5(4.1)

315.

3(5.7)

238.

1(4.3)

401.

7(7.2)

55.7

(1.0)

1034

9.2

(185.8)

7.7

(0.1)

100

1346.0

(8.8)

7328.9

(47.

7)9.

5(0.1)

2248

.2(1

4.6)

3549.6

(23.

1)27

23.1

(17.

7)44

69.4

(29.

1)15

3.6

(1.0)

9479

.6(6

1.7)

9.1

(0.1)

0.75

C5

5.7

(0.3)

6.0

(0.3)

2.4

(0.1)

70.2

(3.8)

95.8

(5.1)

70.8

(3.8)

103.

3(5.5)

18.6

(1.0)

9531.7

(511

.5)

2.3

(0.1)

2055.8

(2.2)

56.7

(2.3)

3.1

(0.1)

258.

1(1

0.3)

352.

1(1

4.0)

308.

5(1

2.3)

425.

8(1

7.0)

25.1

(1.0)

1039

2.1

(414

.6)

2.6

(0.1)

100

1278.9

(21.

2)12

82.7

(21.

3)6.

6(0.1)

2757.5

(45.

8)42

07.0

(69.

9)33

38.1

(55.

5)43

28.8

(71.

9)60

.2(1.0)

9734.6

(161

.7)

3.0

(0.0)

F5

6.9

(0.3)

16.7

(0.6)

4.8

(0.2)

95.0

(3.6)

107.

9(4.0)

98.0

(3.7)

118.

0(4.4)

26.7

(1.0)

1128

4.8

(422.0)

11.5

(0.4)

2057.0

(1.5)

244.

1(6.4)

5.1

(0.1)

333.

4(8.7)

459.

7(1

2.0)

368.

6(9.6)

517.

6(1

3.5)

38.3

(1.0)

1118

5.6

(292.3)

11.3

(0.3)

100

1307.8

(13.

1)59

10.5

(59.

4)9.

7(0.1)

3265.4

(32.

8)38

48.2

(38.

7)40

31.5

(40.

5)45

63.2

(45.

9)99

.5(1.0)

1056

3.6

(106.2)

12.3

(0.1)

G5

6.0

(0.1)

5.6

(0.1)

2.9

(0.1)

56.1

(1.0)

75.5

(1.4)

53.8

(1.0)

88.8

(1.6)

55.3

(1.0)

8612.3

(155.6)

1.5

(0.0)

2056.0

(0.6)

56.7

(0.6)

2.9

(0.0)

239.

9(2.7)

364.

1(4.1)

270.

5(3.0)

403.

7(4.5)

89.1

(1.0)

9264

.3(1

03.9)

1.8

(0.0)

100

1291.9

(4.8)

1279

.1(4.7)

4.2

(0.0)

2332.0

(8.6)

3676.8

(13.

5)25

74.3

(9.5)

4633.4

(17.

1)27

1.4

(1.0)

9082.9

(33.

5)2.

6(0.0)

J5

7.3

(0.2)

16.6

(0.5)

4.8

(0.1)

62.6

(1.9)

92.5

(2.9)

70.1

(2.2)

103.

4(3.2)

32.4

(1.0)

1067

3.4

(329

.2)

5.7

(0.2)

2057.0

(1.1)

308.

1(6.1)

5.3

(0.1)

260.

4(5.1)

382.

4(7.5)

274.

4(5.4)

428.

4(8.4)

50.9

(1.0)

9682.6

(190.2)

7.3

(0.1)

100

1276.6

(11.

0)81

29.0

(70.

0)9.

5(0.1)

2545.1

(21.

9)35

40.6

(30.

5)29

18.5

(25.

1)39

15.4

(33.

7)11

6.1

(1.0)

9530.5

(82.

1)9.

6(0.1)




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